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&ABMAM 




LIBRARY OF CONGRESS. 



ChapHiM Copyright No*. 

Shell. •___$_$ 3 
1 ^oo 



UNITED STATES OF AMERICA. 



^£«>Se£ 



SnJ 








t^e 



WwM 




•7^^- 











\M 




FORMULAS 



IN 



GEAR ING. 



THIRD EDITIO 



-w. ^fctz 



\x 



WITH PRACTICAL SUGGESTIONS 



PROVIDENCE, R. I. 

BROWN & SHARPE MANUFACTURING COMPANY 

1900. 



TWO COPIES RECE1' 

Lftrtry of C§Bgr«8% 
1*« of til 

MAY 7-1880 

fitter of Cspjrtgkt* 

G, /J*s6 2. 



6KCOND COPY, 



38696 



Entered according to Act of Congress, in the year 1900 by 
BRoWX & SHAKPEMFti. CO., 

In the Office "f the Librarian of Congress at Washington. 

Registered at Stationers' Hall. London. Eng. 

All rights reserved. 









PREFACE. 

It is the aim, in the following pages, to condense as much 
as possible the solution of all problems in gearing which in the 
ordinary practice may be met with, to the exclusion of prob- 
lems dealing with transmission of power and strength of 
gearing. The simplest and briefest being the symbolical 
expression, it has, whenever available, been resorted to. The 
mathematics employed are of a simple kind, and will present 
no difficulty to anyone familiar with ordinary Algebra and 
the elements of Trigonometry. 



CONTENTS, 



FORMULAS IN GEARING. 



CHAPTER I. 

Page 
Systems of Gearing .. i 

CHAPTER II. 

Spur Gearing — Formulas — Table of Tooth Parts — Comparative Sizes 

of Gear Teeth 4. 

CHAPTER III, 

Bevel Gears, Axes at Right Angles — Formulas — Bevel Gears, Axes at 
any Angle — Formulas — Undercut in Bevel Gears — Diameter Incre- 
ment — Tables for Angles of Edge and Angles of Face — Tables of 
Natural Lines 1 r 

CHAPTER IV. 

Worm and Worm Wheel, Formulas — Undercut in Worm Wheels — 

Table for gashing Worm Wheels 34 

CHAPTER V. 

Spiral or Screw Gearing — Axes Parallel — Axes at Right Angles — 
Axes at any Angle — General Formulas — Table of Prime Num- 
bers and Factors 40 

CHAPTER VI. 

Internal Gearing — Internal Spur Gearing — Internal Bevel Gears 5S 

CHAPTER VII. 
Gear Patterns 64 

CHAPTER VIII. 
Dimensions and Form for Bevel Gear Cutters 67 

CHAPTER IX. 
Directions for cutting Bevel Gears with Rotary Cutter 70 

CHAPTER X. 
The Indexing of any Whole or Fractional Number 73 

CHAPTER XI. 

The Gearing of Lathes for Screw Cutting — Simple Gearing — Compound 

Gearing — Cutting a Multiple Screw „ 77 



FORMULAS IN GEARING. 



CHAPTER I. 

SYSTEMS OF GEARING. 

(Figs, i, 2.) 

There are in common use two systems of gearing, viz.: the 
involute and the epicycloidal. 

In the involute system the*outlines of the working parts of a 
tooth are single curves, which may be traced by a point in a 
flexible, inextensible cord being unwound from a circular disk 
the circumference of which is called the base circle, the disk 
being concentric with the pitch circle of the gear. 




Fig. 1. 



In Fig. i the two base circles are represented as tangent to 
the line P P. This line (P P) is variously called " the line of 
pressure," " the line of contact," or " the line of action." 



2 BROWN i 5HARPE MFG. CO. 

In our practice this is drawn so as to make with a normal 
to the center line (O O') 14^ ~. or with the center line 75 : _ ". 

The rack of this system has teeth with straight sides, the two 
sides of a tooth making, together, an angle of 29° (twice 

t/ o. 

This applies to gears having 30 teeth or more. For gears 
having less than 30 teeth special rules are followed, which are 
explained in our " Practical Treatise on Gearing." 




Fig. 2. 



In epicydoiaaL or double-curve teeth, the formation of the 
curve changes at the pitch circle. The outline of the faces of 
epicycloidal teeth may be traced by a point in a circle rolling 
on the outside of pitch circle of a gear, and the.flanks by a point 
in a circle rolling on the inside of the pitch circle. The faces 
of one gear must be traced by the same circle that traces the 
flanks of the engaging gear. 

In our practice the diameter of the rolling or describing 
circle is equal to the radius of a 15-tooth gear of the pitch 
required ; this is the base of the system. The same describing 
circle being used for all gears of the same pitch. 



PROVIDENCE, R. f. 3 

The teeth of the rack of this system have double curves, 
which may be traced by the base circle rolling alternately on 
each side of the pitch line. 

Ail advantage of the involute over the epicycloidal tooth is, 
that in action gears having involute teeth may be separated a 
little from their normal positions without interfering with the 
angular velocity, which is not possible in any other kind of 
tooth. 

The obliquity of action is sometimes urged as an objection 
to involute teeth, but a full consideration of the subject will 
show that the importance of this has been greatly over-esti- 
mated. 

The tooth dimensions for both the involute and epicycloidal 
gears may be calcu' j.ted from the formulas in Chapter II. 



BROWN & SHARPE MFG. CO. 



CHAPTER. II. 



SPUR GEARING. 

(Figs. % 4.) 

Two spur gears in action are comparable to two correspond- 
ing plain rollers whose surfaces are in contact, these surfaces 
representing the pitch circles of the gears. 

Pitch of Gears. 

For convenience of expression the pitch of gears may be 
stated as follows : 

Circular pitch is the distance from the center of one tooth to 
the center of the next tooth, measured on the pitch line. 

Diametral pitch is the number of teeth in a gear per inch of 
pitch diameter. That is, a gear that has, say, six teeth for each 
inch in pitch diameter is six diameuai pitch, or, as the expres- 
sion is universally abbreviated, it is "six pitch." This is by 
far the most convenient way of expressing the relation of 
diameter to number of teeth. 

Module is the pitch diameter of a gear divided by the 
number of teeth. 

Chordal pitch is a term but little employed. It is the dis- 
tance from center to center of two adjacent teeth measured in 
a straight line. 




Fig. 3. 



r~t—*i 



PROVIDENCE, R. I. 



FORMULAS. 



N = number of teeth. 

s — addendum and module. 

/ — thickness of tooth on pitch line, 

/= clearance at bottom of tooth, 
D" = working depth of tooth. 
D' r + / = whole depth of tqcch. 

d = pitch diameter. 
d' = outside diameter. 
P' = circular pitch. 
P^ = chord pitch. 
P = diametral pitch, 
C = center distance. 



p _ N + 2 
d' 

71 



P =P' 

P' = ? 



P 71 



= .318.3 P' 



d _ d' 



N N + 2 

/=^P'=:_^L_ 
2 2 P 

y = — ^ 
10 



+/= t( i + 3=- 568sP 



s 
P c = d sin 



N 
360 5 ' 1 ^ 



(J . P r 

P' = dn where sin d — 



/ N 
P 



d' = d + 2 s 

, NP' 
# = 

7t 



BROWN & SHARPE MFG. CO. 



GEAR "WHEELS. 



TABLE OF TOOTH PARTS CIRCULAR PITCH IN FIRST COLUMN. 





Threads or 

Teeth per inch 

Linear . 


Diametral 
Pitch. 


Thickness of 

Tooth on 
Pitch Line. 


Addendum 
and Modnle. 


Working Depth 
of Tooth. 


Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


Width of 

Thread-Tool 

at End. 


Width of 
Thread at Top. 


p' 


l" 
p< 


P 


f 


8 


D" 


•+/ 


D"+ / 


P'X.31 


P'X.335 


2 


i 

2 


1.5708 


1.0000 


.6366 


1.2732 


.7366 


1.3732 


.6200 


.6700 


If 


8 
15 


1.6755 


.9375 


.5968 


1.1937 


.6906 


1.2874 


.5813 


.6281 


14 


_4_ 

7 


1.7952 


.8750 


.5570 


1.1141 


.6445 


1.2016 


.5425 


.5863 


1+ 


8 
13 


1.9333 


.8125 


.5173 


1.0345 


.5985 


1.1158 


.5038 


.5444 


ll 


2 
3 


2.0944 


.7500 


.4775 


.9549 


.5525 


1.0299 


.4650 


.5025 


it 


16 
23 


2.1855 


.7187 


.4576 


.9151 


.5294 


.9870 


.4456 


.4816 


If 


8 
11 


2.2848 


.6875 


.4377 


.8754 


.5064 


.9441 


.4262 


.4606 


if 


3 
J. 


2.3562 


.6666 


.4244 


.8488 


.4910 


.9154 


.4133 


.4466 


If 


16 
21 


2.3936 


.6562 


.4178 


.8356 


.4834 


.9012 


.4069) .4397 


H 


4 

5 


2.5133 


.6250 


.3979 


.7958 


.4604 


.8583 


.38751 .4188 


It 


16 
19 


2.6456 


.5937 


.3780 


.7560 


.4374 


.8156 


.3681! .3978 


If 


8 
9 


2.7925 


.5625 


.3581 


.7162 


.4143 


.7724 |.3488| .3769 


If 


16 
17 


2.9568 


.5312 


.3382 


.6764 


.3913 


.7295 


.32941 .3559 


1 


1 


3.1416 


.5000 


.3183 


.6366 


.3683 


.6866 


.3100 


.3350 


15 
16 


li 


3.3510 


.4687 


.2984 


.5968 


.3453 


.6437 


.2906 


.3141 


8 


1+ 


3.5904 


.4375 


.2785 


.5570 


.3223 


.6007 


.2713 


.2931 


13 

16 


li 13.8666 


.4062 


.2586 


^,5173 


.2993 


.5579 


.2519 


.2722 


4' 
5 


1-f 13.9270 


.4000 


.2546 


.5092 


.2946 


.5492 


.2480 


.2680 


3 

4 


li 


4.1888 


.3750 


.2387 


.4775 


.2762 


.5150 


.2325 


.2513 


11 
16 


1* 


4.5696 


.3437 


.2189 


.4377 


.2532 1 .4720 


.2131 


.2303 


2 
3 


1+ 


4.7124 


.3333 


.2122 


.4244 


.2455; .4577 


.2066 


.2233 


5 
8 


1 5 


5.0265 


.3125 


.1989 


.3979 


.2301 


.4291 


.1938| .2094 


3 
5 


-L 3 


5.2360 


.3000 


.1910 .3820 


.2210 


.4120 


.1860 


.2010 


4 

; 


If 


5.4978 


.2857 


.1819 


.3638 


.2105 


.3923 


.1771 .1914 


9 

16 


If 


5.5851 


.2812 11790 


.3581 


.2071 .3862 


.1744 .1884 

- 



PROVIDENCE, R. I. 



TABLE OF TOOTH ¥ ARTS.— Continued. 



CIRCULAR PITCH IN FIRST COLUMN. 



f-l 

1 — 1 <-i 

a -£ 

^ IT 1 


Threads or 

Teeth per inch 

Linear. 


Diametral 
Pitch. 


Thickness of 

Tooth on 
Pitch Line. 


Addendum 
and Module. 


Working Deptl 
of Tooth." 


Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


Width of 

Thread-Tool 

at End. 


Width of 
Thread at Top. 


P' 


i" 
p' 


P 


t 


S fc 


D" 


«+/ 


J)\f. 


PX.31 


PX.335 


1 

2 


2 


6.2832 


.2500 


.1592 


.3183 


.1842 


.3433 


.1550 


.1675 


"7T 


2t 


7.0685 


.2222 


.1415 


.2830 


.1637 


.3052 


.1378 


.1489 


7 
1G 


2f 


7.1808 


.2187 


.1393 


.2785 


.1611 


.3003 


.1356 


.1466 


_3_ 


2i 


7.3304 


.2143 


.1364 


.2728 


.1578 


.2942 


.1328 


.1436 


2 
5 


^2 


7.8540 


.2000 


.1273 


.2546 


.1473 


.2746 


.1240 


.1340 


3 
8 


2f 


8.3776 


.1875 


.1194 


.2387 


.1381 


.2575 


.1163 


.1256 


4 
11 


2f 


8.6394 


.1818 


.1158 


.2310 


.1340 


.2498 


.1127 


.1218 


1 
3 


3 


9.4248 


.1666 


.1061 


.2122 


.1228 


.2289 


.1033 


.1117 


5 
10 


31 


10.0531 


.1562 


.0995 


.1989 


.1151 


.2146 


.0969 


.1047 


3 
10 


31 


10.4719 


.1500 


.0955 


.1910 


.1105 


.2060 


.0930 


.1005 


2 

7 


3- 


10.9956 


.1429 


.0909 


.1819 


.1052 


.1962 


.0886 


.0957 


1 
i 


4 


12.5664 


.1250 


.0796 


.1591 


.0921 


.1716 


.0775 


.0838 


2 
9 


4- 

^ 2 


14.1372 


.1111 


.0707 


.1415 


.0818 


.1526 


.0689 


.0744 


1 

5 


5 


15.7080 


.1000 


.0637 


.1273 


.0737 


.1373 


.0620 


.0670 


3 
1G 


oy 


16.7552 


.0937 


.0597 


.1194 


.0690 


.1287 


.0581 


.0628 


2 
11 


5f 


17.2788 


.0909 


.0579 


.1158 


.0670 


.1249 


.0564 


.0609 


1 

6 


6 


18.8496 


.0833 


.0531 


.1061 


.0614 


.1144 


.0517 


.0558 


2 
13 


6+ 


20.4203 


.0769 


.0489 


.0978 


.0566 


.1055 


.0477 


.0515 


1 

7 


7 


21.9911 


.0714 


.0455 


.0910 


.0526 


.0981 


.0443 


.0479 


2 
15 


7- 
< 2 


23.5619 


.0666 


.0425 


.0850 


.0492 


.0917 


.0414 


.0446 


1 
8 


8 


25.1327 


,0625 


.0398 


.0796 


.0460 


.0858 


.0388 


.0419 


1 
9 


9 


28.2743 


.0555 


.0354 


.0707 


.0409 


.0763 


.0344 


.0372 


1 
10 


10 


31.4159 


.0500 


.0318 


.0637 


.0368 


.0687 


.0310 


.0335 


1 
16 


16 


50.2655 


.0312 


.0199 


.0398 


.0230 


.0429 


.0194 


.0209 


1 

20 


20 


62.8318 


.0250 


.0159 


.0318 


.0184 


.0343 


.0155 


.0167 



BROWN & SHARPE MFG. CO. 



GEAR WHEELS. 



TABLE OF TOOTH PARTS DIAMETRAL PITCH IN FIRST COLUMN. 



'3 

5^ 


53 ■ 

5^ 


Thickness 
of Tooth on 
Pitch Line. 


Addendum 
and Module. 


5 c 


Depth of Space 

below 

Pitch Line. 


Whole Depth 
of Tooth. 


P 


P< 


t 


s 


D" 


•+/■ 


D"+/. 

4.3142 


1 
2 


6.2832 


3.1416 


2.0000 


4.0000 


2.3142 


2. 
4 


4.1888 


2 , 0944 


1.3333 


2.6666 


1.5428 


2.8761 


1 


3.1416 


1 . 5708 


1.0000 


2.0000 


1.1571 


2.1571 


ii 


2.5133 


1.2566 


.8000 


1.6000 


.9257 


1.7257 


i* 


2.0944 


1.0472 


.6666 


1.3333 


.7714 


1.4381 


if 


1.7952 


.8976 


.5714 


1.1429 


.6612 


1.2326 


2 


1.5708 


.7854 


.5000 


1.0000 


.5785 


1.0785 


2J 


1.3963 


.6981 


.4444 


.8888 


.5143 


.9587 


?4 


1.2566 


.6283 


.4000 


.8000 


.4628 


.8628 


2f 


1.1424 


.5712 


.3636 


.7273 


.4208 


.7814 


3 


1.0472 


.5236 


.3333 


.6666 


.3857 


.7190 


3i 


.8976 


.4488 


.2857 


.5714 


.3306 


.6163 


4 


.7854 


.3927 


.2500 


.5000 


.2893 


.5393 


5 


.6283 


.3142 


.2000 


.4000 


.2314 


.4314 


G 


.5236 


.2618 


.1666 


.3333 


.1928 


.3595 


7 


.4488 


.2244 


.1429 


.2857 


.1653 


.3081 


8 


.3927 


.1963 


.1250 


.2500 


.1446 


.2696 


9 


.3491 


.1745 


.1111 


.2222 


.1286 


.2397 


10 


.3142 


.1571 


.1000 


.2000 


.1157 


.2157 


11 


.2856 


.1428 


.0909 


.1818 


.1052 


.1961 


12 


.2618 


.1309 


0833 


.1666 


.0964 


.1798 


13 


.2417 


.1208 


.0769 


.1538 


.0890 


.1659 


14 

_ 


. 2244 


.1122 


.0714 


.1429 


.0826 


.1541 



PROVIDENCE, R. I. 



TABLE OF TOOTH PABTS— Continued 



DIAMETRAL PITCH IN FIRST COLUMN. 



~3 
Q 


£3 ,- 


Thickness 
of Tooth on 
Pitch Line. 


a. ° 


a, 


Depth of Space 

below 

Pitch Line. 


-4-> 

a-; 

Q ° 

M o 

2<~ 


P. 


P'. 


t. 


s. 


D". 


.0771 


D"+/. 

.1438 


15 


.2094 


.1047 


.0666 


.1333 


16 


.1963 


.0982 


.0625 


.1250 


.0723 


.1348 


17 


.1848 


.0924 


.05^8 


.1176 


.0681 


.1269 


18 


.1745 


.0873 


.0555 


.1111 


.0643 


.1198 


19 


.1653 


.0827 


.0526 


.1053 


.0609 


.1135 


20 


.1571 


.0785 


.0500 


.1000 


.0579 


.1079 


22 


.1428 


.0714 


.0455 


.0909 


.0526 


.0980 


24 


.1309 


.0654 


.0417 


.0833 


.0482 


.0898 


26 


.1208 


.0604 


.0385 


.0769 


.0445 


.0829 


28 


.1122 


.0561 


.0357 


.0714 


.0413 


.0770 


30 


.1047 


.0524 


.0333 


.0666 


.0386 


.0719 


32 


.0982 


.0491 


.0312 


.0625 


.0362 


.0674 


34 


.0924 


.0462 


.0294 


.0588 


.0340 


.0634 


36 


.0873 


.0436 


.0278 


.0555 


.0321 


.0599 


38 


.0827 , 


.0413 


.0263 


.0526 


.0304 


.0568 


40 


.0785 


.0393 


.0250 


.0500 


.0289 


.0539 


42 


.0748 


.0374 


.0238 


.0476 


.0275 


.0514 


44 


.0714 


.0357 


.0227 


.0455 


.0263 


.0490 


46 


.0683 


.0341 


.0217 


.0435 


.0252 


.0469 


48 


.0654 


.0327 


.0208 


.0417 


.0241 


.0449 


50 . 


.0628 


.0314 


.0200 


.0400 


.0231 


.0431 


56 


.0561 


.0280 


.0178 


.0357 


.0207 


.0385 


60 


.0524 


.0262 


.0166 


.0333 


.0193 


. 0360 



IO 



BROWN & SHARPE MFG. CO. 



Comparative Sizes of Gear Teeth. 
Involute. 





*<J P 



18 P 



16 P 



14 P 



12 P 



7 P 




IO P 




S P 



Fig. 4. 




y p 



PROVIDENCE, R. I. 



II 



CHAPTER. III. 



BEVEL GEARS.— AXES AT RIGHT ANGLES, 



(Fig. 5.) 




12 BROWN 6c SHARPE MFG. CO. 



FORMULAS. 

?j a = I Number of teeth \ gea . r ' 
jN & — ) I pinion 

P = diametral pitch. 

P' = circular pitch. 

a a = ) center angle = angle of edge j gear. 
w b = \ or pitch angie ( pinion. 

ft = angle of top. 

ft' = angle of bottom. 

*" ~ - angfle of face \ °. . r ' 
gb = ) & ( pinion. 

%a =1 4-4.- i f gear. 

7 v cutting: angle s . • 

n b = j & & ( pinion. 

A = apex distance from pitch circle. 

A' = apex distance from large bottom of tooth. 

d= pitch diameter. 

d' = outside diameter. 

s — addendum and module. 

t = thickness of tooth at pitch line. 

f = clearance at bottom of tooth. 

D" = working depth of tooth. 

D" -f f = whole depth of tooth. 

2 a = diameter increment. 

b — distance from top of tooth to plane of pitch cirde. 

F = width of face. 



PROVIDENCE, R. I. 13 



N a . N b 
tan a a = — ; tan a b = - — ; 

N 6 b N a 

a 2 sin a s 

tan a = ; or tan B = — • 

' N ' ' A 

tan p = sina ( 2 + ^) = 2 -3 J 4 sin a t g = s_ + f 
N N ' A ' 

ga - 9°° ~ (ot a + 0) ; g b = 90 - (» 6 + /i) 
h = a— ft' (See Note, page 6 p. ) 

2 



A ^@ 2 + (S) : 



A 


— 


N 






2 P sin 


« 


A' 


— 


A 






cos p' 




A 


= 


\*' 






sin (a + 









N 





A' N 



2 P sin a cos /ft' 

cos /i 



p — IN 




2 A sin c* 




, N N P' 
<? — or — 

P 71 


d' = d -f 2 * 


2 a = 2 s cos a 


(See page 20.) 



, [ a for orear = b for pinion 

# = a tan « •< . & . . z r 

( # for pinion = b tor gear 

p = _5_ p'= n 

P' P 

^ = — = — = .3183 P' s = Atan/i 

P 7T 

j- + / - . 3685 P' s + / = A tan /?' 

2 2 P 



F 



i + - or = 2 P' to 3 P' 



Note. — Formulas containing notations without the designating- letters a and b 
apply equally to either gear or pinion. If wanted for one or the other, the respective 
letters are simply attached. 



14 



BROWN & SHARPE MFG. CO. 



BEVEL GEARS WITH AXES AT ANY ANGLE. 




Fig. (>. 



PROVIDENCE, R. I, J 5 



FORMULAS. 



C = angle formed by axes of gears. 
<- number of teeth J =» 



xt >• numoer 01 teem -\ *-*. . 

N b = j ( pinion. 

P = diametral pitch. 

P' = circular pitch. 

a ~ [• angle of edge = pitch angle \ °. . " 
a b = j & e> r ( pinion. 

(5 = angle of top. 

fi' = angle of bottom. 

* a ~~ f- angle of face \ **. • ' 
gb = j s ( pinion. 

/* tt = ) , ( gear. 

7 a y cutting angle ■< & . . 
n b = j & & ( pinion. 

A = apex distance from pitch circle. 
A' = apex distance from large bottom of tooth. 

d= pitch diameter. 
d' = outside diameter. 
2 « = diameter increment. 

b = distance from top of tooth to plane of pitch circle. 



Note. — The formulas for tooth parts as given on page 5 apply equally to these 
cases. 



l6 BROWN & SHARPE MFG. CO. 



tan a„ 




sin C 


~ N ft 


+ cos C 
sin C 




" X., 


-f cos C 



Ni, 
; or cot a a = — — : — — + cot C 

N„ sin C 



; or cot <r,, = — — : — — -+- cot C 
X i sin L 



N,, 



Note. — The above formulas are correct only for values of C less than 90 . 
If C is greater than 90 , consult page 18. 



2 sin Ol s 

tan P = — -- — ; or tan p == - T - ; 
X A 

tan /r = Bin « (»+£) = 2 - 3 H sin ^ = ,+,/ 

XX A 

o-, ( — g ° — ( a'a -f- fi ) for Cases I and II. 

ga — fi, for Case III. 

g a = 90 — (a'a — /?) for Case IV. 

^ = 9 o°— (flf 4 + fi) 

// = a — /5' (See page 6p. ) 

A = p * 

2 P sm a? 

a-= A 



Cos ft' 

N X P' 

</ = — or = - - 

P 7T 

., { for Cases I and II, 

a = a -\- 2 a 

I and pinions in Cases III and IV. 

a" — a 7 , for gear in Case III. 

d' = d — 2 a, for srear in Case IV. 

2 # = 5 cos a 

b = s sin « 

Note. — Formulas containing notations without the designating letters a and 
apply equally to either gear or pinion. If wanted for one or the other, the 
respective letters are simply attached. 



PROVIDENCE, R. I. 



i; 




l8 BROWN & SHARPE MFG. CO. 



The formulas given for a a and a h (when C. X„ and N 6 are 

known) undergo some modifications for values of C greater 
than 90 . 

For bevel gears at any angle but 90 we may distinguish 
four cases ; C, X„, N 6 being given. 

/. Case. See pages 14 and 16. 

II. Case. C is greater than 90 . 

sin (180 — C) sin (180 — C 

tan a a =z — 5 '— ; tan a b — 



v ' u v 

— 6 — cos (180-C) _ a -cos ( 1S0-C, 

N N, 

*■ * a " > ;» 

///. Case. a a — 90 ; a b = C — 90 
IV. Case. 

sin E sin E 

tan (Y„ — _ ; tan a b = 



cos E — ± - b — — cos E 

N d X, 

For an example to apply to Case III., the following condi- 
tion must be fulfilled : 

■ N a sin (C — 90°) = X,, 

To distinguish whether a given example belongs to Case II. 
or case IV., we are guided by the following condition : 

x XT . //-, \ \ smaller than X,,, we have Case II. 

Is : X„ sin (C — 90 ) \ 7 ^ u XT 6 ' , TTr 

y ' ( larger than N 6j we have case IV. 



PROVIDENCE, R. I. 19 



UNDERCUT IN BEVEL GEARS. 

By undercut in gears is understood a special formation of 
the tooth, which may be explained by saying that the elements 
of the tooth below the pitch line are nearer the center line of 
the tooth than those on the pitch line. Such a tooth outline is 
to be found only in gears with few teeth. In a pair of bevel 
gears where the pinion is low-numbered and the ratio high, we 
are apt to have undercut. For a pair of running gears this 
condition presents no objection. Should, however, these gears 
be intended as patterns to cast from, they would be found use- 
less, from the fact that they would not draw out of the sand. 
We have stated on page 2 (see Fig. 1) that the base of our 
involute system is the 14^° pressure angle. If a pair of bevel 
gears with teeth constructed on this basis have undercut, we 
can nearly eliminate the undercut — and for the practical work- 
ing this is quite sufficient— by taking as a basis for the con- 
struction of the tooth outline a pressure angle of 20 . 

The question now is : When do we, and when do we not 
have undercut ? Let there be : 

N = number of teeth in gear. 
7/ = number of teeth in pinion. 



/ 
n V N 2 + ;/ 2 



= P 



N 
where we have undercut for/ less than 30. 

This formula is strictly correct for epicycloidal gears only. 
It is, however, used as a safe and efficient approximation for 
the involute system. 



20 



BROWN & SHARPE MFG. CO. 



DIAMETER INCREMENT. 

2 a. 

Rule. — The ratio being given or determined, to find the outside diameter 
divide figures given in table for large and small gear by pitch (P) and add 
quotient to pitch diameter. 







GEARS. 






GEARS. 




GEARS. 


RATTO 






RATTO 






RATIO . 






1:1 


Large 


Small 






Large 


Small 




Large 


Small 


1.00 


1.41 


1.41 


1.65 




1.05 


1.70 


4.40 




.45 


1M 


1.05 




1.37 


1.42 


1.67 


5:3 


1.03 


1.72 


4 50 


9:2 


.44 


1.95 


1.07 




1.36 


1.43 


1.70 




1.01 


1.73 


4.60 




.42 


1 95 


1.10 




1.35 


1.44 


1.75 


7:4 


.99 


1.74 


4.80 




.41 


1.96 


1.11 


10:9 


1.34 


1.46 


1.80 


9:5 


.97 


1.75 


5.00 


5:1 


.39 


1.96 


1.12 




1.33 


1.46 


1.85 




.95 


1.76 


5.20 




.38 


1.96 


1.13 


9:8 


1.33 


1.47 


1.90 




.93 


1.77 


5.40 




.37 


1.96 


1.14 


8:7 


1.32 


1.49 


1.95 




.91 


1.78 


5.60 




.36 


1.97 


1.15 




1.31 


1.50 


2.00 


2:1 


.89 


1.79 


5.80 




.34 


1.97 


1.16 




1.30 


1.51 


2.10 




.87 


1.80 


6.00 


6:1 


.33 


1.97 


1.17 


7:6 


1.30 


1.52 


2.20 




.84 


1.81 


6.20 




.32 


1 97 


1.18 




1.29 


1.53 


2 25 


9:4 


.82 


1.82 


6.40 




.31 


1.97 


1.19 




1.28 


1.53 


2.30 




.80 


1.83 


6.60 




.30 


1 97 


1.20 


6:5 


1.28 


1.54 


2.33 


7:3 


.78 


1.84 


6.80 




.29 


1 98 


1.23 




1.27 


1.55 


2.40 




.76 


1.85 


7 00 


7:1 


.28 


1.98 


1.25 


5:4 


1.25 


1.56 


2.50 


5:2 


.75 


1.86 


7.20 




.27 


1.98 


1.27 




1.25 


1.57 


2.60 




.73 


1.86 


7.40 




.27 


1 98 


1.29 


9:7 


1.24 


1.58 


2.67 


8:3 


.71 


1.87 


7.60 




.26 


1 98 


1.30 




1.22 


1.59 


2.70 




.69 


1.87 


7 80 




.26 


1.98 


1.33 


4:3 


1.20 


1.60 


2.80 




.67 


1.88 


8 00 


8:1 


.25 


1.98 


1.35 




1.18 


1.61 


2.90 




.65 


1.89 


8.20 




.24 


1.98 


1 37 




1.17 


1.61 


3.00 


3:1 


.63 


1.91 


8 40 




.24 


1.98 


1.40 


7:5 


1.16 


1.62 


3.20 




.60 


1.92 


8.60 




.23 


1.98 


1.43 


10:7 


1.15 


1.63 


3.33 




.58 


1.92 


8.80 




.23 


1.98 


1.45 




1.13 


1.65 


3.40 




.56 


1 92 


9.00 


9:1 


.22 


1.99 


1.50 


3:2 


1.11 


1.66 


3.50 


7:2 


.54 


1.93 


9.20 




.22 


1.99 


1.53 




1.10 


1 67 


3.60 




.52 


1 93 


9.40 




.21 


1.99 


1 55 




1.09 


1 67 


3.80 




.50 


1.94 


9.60 




.21 


2.00 


1 . 58 




1.08 


1.68 


4.00 


4:1 


.49 


1.94 


9.80 




.20 ; 2.00 


1.60 


8:5 


1.07 


1.68 

l 


4.20 




.47 


1.94 


10.00 


10:1 


.20 


2 00 



Note. — To be used only for bevel gears with axes at right angle. 



PROVIDENCE, R. I. 21 



TABLES FOR ANGLES OF EDGE AND ANGLES 

OF FACE. 

The following four tables have been computed for the 
convenience in calculating datas for bevel gears with axes at 
right angle. They do not hold good for bevel gears with axes 
at any other angle. 

To use the tables the number of teeth in gear and pinion 
must be known. 

Having located the number of teeth in the gear on the 
horizontal line of figures at the top of the table, and the num- 
ber of teeth in the pinion on the vertical line of figures on the 
left hand side, we follow the two columns to the square formed 
by their intersections. 

The two angles found in the same square are the respective 
angles for gear and pinion. The tables are so arranged that 
the angle belonging to the gear is always placed above the 
angle for the pinion. 



22 



BROWN & SHARPE MFG. CO. 



TABLE i 

Angle of Edge, 
gear. 







41 


40 


39 


38 


37 


36 


35 


34 


33 


32 


3! 


30 


29 


28 


27 




12 


73V 

16*19 


73°i8 

16*42 


72V 
17V 


72*28 
17*32 


72*2 
17*58 


71 V 

18*26 


71*5' 
18*55 


70V 

19*25 


70 V 
19V 


69*26 

20V 


68°50' 
2l*o 


68*ie' 
2 1 V 


67°3i' 
22 V 


66V 
23*12 


66*2' 

23*58 




13 


72 W 

17*35 


71 59 
18* \ 


71*34 

18*26 


71 V 
18*53' 


70*39' 

I9*ei' 


70*9' 
19V 


69*37' 
20*23 


69*5 

20*55 


68 v 30 

21*30 


67*53 

22*7' 


67'is 1 

22*45 


66 V 

23*26 


65V 
24's' 


65*6 

24*54 


64V 

25*43 




14 


71V 

18*51' 


70*43 
19*17 


70*is 

19*45 


69V 
20*14 


69*16 

20*44 


68*45 

2lV 


68*12 

21*48 


67V 

22*23 


67*0 
23°o' 


66*23 
23*37 


6sV 
24*18 


64V 
25* 1' 


64V 
25*46 


63 V 
26V 


62*36 
27*24 




15 


69V 
20*6' 


69*86 
20*34 


68*58 
21*2 


68£8 
21*3* 


67"V 
22*4 


67*23 

22*3?' 


66*48 

23*2 


66'i2 

23*48 


65*33 
24*27 


64*53' 

25*7' 


64V 
25°So 


63*26 
26*34 


62*39 
27*21 


61*49 
28*i' 


60V 
29*3 




16 


68V 
2I*W 


68'is' 

8148 


67 V 

e2V 


67*10 
22*so 


66*37 
23*23' 


66V 
23°S8 


65*26' 
24*34 


64*48 
25*12 


64*8' 

25*58 


63*2& 

26*34 


62°V 
27*« 


6lV 
28°*' 


61 V 

28*53 


60°is 

29*45 


59*2i' 
30*33 




17 


67*29' 

22*3 1 


66*58 

23*2 


66*27 

23*33 


65*54 
24*6 


65*19 
84V 


64*43 

25*7 


64*6 
25°S4 


63 26 

26*34 


62V 
27°is' 


62* r 

27*59 


61*15 
28*45 


60V 

29*32 


59*37 

30V 


58V 

31*16 


57 V 
32*12 




18 


66V 

2348' 


6546 

24°i4 


65*14 
24*46 


64*39 

25*21 


64*4' 
25V 


63V 
26V 


62*47 
27°<i 


62*6 

27*54 


61*5 

28V 


60*38 

29*22 


59 V 
30*9 


59*2 

30°se 


58*id 

31*50 


57V 

32*44 


56V' 
33*4i' 




19 


65*8 
24°s* 


64*3* 
25*24 


64V 
25*58 


63*26 

26°B4 


62*49 

e7°u' 


62*io 
27*50 


61*10 
28*3© 


60*48 
29*12 


60*4 

29*56 


59*18 

30*42 


58*30 

31*30 


57°39' 
32*ei' 


56i6 
33*4 


55*Si' 
34*9' 


54*52 
35*8' 




20 


64"V 

26*o' 


63*26 
26*34 


62V 
27*9 


62*14 

27*46' 


6l°37 
28*23 


60*57 
89** 


S0°is 

29*45 


59V 
30*28 


58V' 

31*13' 


58V 
32V 


57V 
32*50 


56° 19 
33*41' 


55*24 
34*36 


54°28 

35*32 


53*28 

36*32' 




21 


62V 
27*7' 


62* » 
27*4*: 


6I*« 

28*19 


61*4' 
28*56 


60°25 

29°3s' 


59« 
30*is' 


59° Z 

30*58 


58*8 

31*42 


57*32 

32*28 


5€V 
33*iV 


55*53' 
34*7' 


55°o' 
35° 0' 


54*5' 

35*55 


53*7' 

36*53 


52*8 
37a 




22 


61*47 
28°i3 


61" n 

28*49 


60°34 
29*26 


59*56 
30V 


59°iS 

30*45 


58*34 
3 1*26 


575i' 
32*9' 


57*6' 
32*54 


56°i9' 
33*4)' 


55*29 

34*3i' 


54*38 
35*22 


53*45 

36*ts' 


52*43 

37*11' 


51*50 

38°id 


50*49' 

39*11' 




23 


60V 
29°i8 


60*6 
29°5* 


59*28 
30°32 


5849 

31*11* 


58*3 

31*52 


57*25 
32*3S 


S6*V 
33* » 


55*55 

34*s' 


55*7' 

34*53 


54*» 

35*42 


53*26 
36*V 


52*3.' 
37» 


51*35 

38°25 


50V 

39*24 


43°V 

40°26 




24 


59*39 

30*2t' 


59*2' 
30*S8 


58*23 
3 1 *37 


57*44 

32*16 


57V 

32*58 


56*19 
33V 


55*33 
34*27' 


54V 
35*3 


53V 

3&Y 


53*7' 
36*53 


52*15 

37*45 


5I°20' 

38*40' 


50*23 
39*37 


4aV" 

40*36 


48*22 
41*38' 




25 


S8ss 
31*22 


58V 
32*0' 


5720 
3240 


56V 
33*eo 


55V 
34*3' 


S5°i3' 

34*47 


54*28' 
35*32' 


53V 

36*20 


52*si' 
37V 


52V 
38V 


51*7' 
38°S3 


5oV 
39°48' 


49*14 
40*46 


48*14 
41*46 


47V' 
42*48 


'A 


26 


57*37' 

32*23' 


56*58 
33*e' 


5€'i9 
33°4i' 


55*37 

34*23 


54*54 
35*6 


54° id 
35*5d 


53*24 

36*3* 


52*36 
37*24 


51*46 

38*14 


50S4 
39*6 


50° r 
39°S3 


49*5 
40*55 


48*7 
41*53 


47'7' 
42*53 


46*5 
43*55 


/- 


27 


56*38 
33*22 


55*59 
34*.' 


ss** 

34°42 


54*36 
35*24 


53*53 
36*7 


53*7 
36*53 


52V 
37*39' 


51*33 
38*27 


50V 

39*17 


49°SI 
40*9 


4857 
41° 3' 


48V 
42°o 


47*3' 
42V 


46°2 
43°58 


45* 




28 


55*4* 
34w 


55*o 
35*o' 


54° 19 
35*41 


53*37 
36*23 


52*53 
37*7' 


52V 
37*S2 


51*20 
38*40 


50*32 

39*28 


49 V 
40 19 


48*48 

41*12 


47*55 
42*5 


46*58 
43*2 


46*o' 
44°o' 


45* 






29 


54V 
35*i6 


54*3' 
35*S7 


53V 

36*38' 


S2 C 33' 

37V 


51*55 
38° S 


51*9 
38°5i 


5021 
39*39 


49*32' 
40*28 


48°4i' 

41*19 


47V 
42* w' 


46 V 
43*6 - 


4S , "58' 
44*2' 


45* 








30 


S3*« 

36*12 


53*7 

36*53' 


52*16 

37*34 


51*42 

38°ie' 


50°S8 

39*2 


50*ie 

39*48 


49*24 

40*36 


48*35 

41*25 


4743 
42*17 


46*5 1 
43° 9 


45*56 
44V 


45° 








31 


52*54 

37V 


52*3 
37*47 


SlV 

58w 


50*48' 
39*2 


50V 

39°S8 


49°i6 

40*44 


48*28 
41 V 


47*39 
42*2f 


46*47 

43*13 


45°S4 
44*6 


45* 








32 


52V 
37ss 


51*20 

38°4o 


50*38 
39*22 


49*54 
40*6 


49*9 
40V 


48*22 
41 V 


47V 

42*26 


46 V 
43*16 


45V 
44*7' 


45° 








33 


Sl'io 
38 so 


5029 

39*31 


49*46 

40*4 


43*2' 

40*56 


48*16 

41*44 


47*29 
42*2 1' 


4641 
43°i» 


455» 
44*9 


45° 








34 


50*20 
3940 


49*38 
40°« 


48V 
41*5' 


48V 
4lV 


47*25' 
42*35 


46*38 
43*22 


4S"so 
44*i o' 


45* 








35 


49*3i 
40a 


48*48 

4l*ie 


46*5 
4I°55 


47°n 

42*39 


46°35 
43*25 


45*48 

44*« 


4-5° 








36 


48*43 
41*17 


48°0 
42*o 


47o' 

4243 


46*33' 
43*27 


45*47 

44*13 


45' 








37 


47 56 

42V 


47°i4 
42*46 


46V 
43*30 


454* 

44V 


45° 








38 


47'io 
42*so 


46*a 

43*32 


45V 

44*15 


45* 








39 


46V 
43V 


4543 

44°>7 


45' 








40 


45*42 

44*18 


45° 








41 


45* 













PROVIDENCE, R, I. 



2 3 



TABLE i. — {Continued}) 



Angle of Edge. 



gear. 





26 


25 


24 


i23 


22 


21 


20 


19 


18 


17 


16 


15 


14 


13 


12 




12 


65°i4 

24*46 


64*22 
Z5°38 


63*26 
26*34 


6227 
27*33 


61*23 
28*37 


60°i5 
29*45 


59*2 

30*58 


574* 
32*16 


56*19 
33*4i' 


54*47 

35*13 


53*7' 

36*53 


5 1 °20 
38*40 


49*24 
40*36 


47*17 

42*43 


45* 




13 


63*26 

26*34' 


62*31 

27*29 


61 °33 
28*27' 


60V 
ib'ea 


59*25 

30*35 


58*rt 

3 1 °46 


56*58 
33° z 


55*37 
34*23 


54"io 
35*50' 


52*36 
37*24 


50*54 

39°6 


49*5 

40SS 


47*7 

42*53 


45* 






14 


61*42 

2 8° 8 


6045 
29° s 


59°4S 
30*15 


58*40 

3 1 °20 


57 32 
32*28 


56*19 
33°4i' 


55° 0' 
35*0 


53*37 

36*23 


52*8 

37*52 


50*32 
39*26 


48 48 
4I°I2 


46*58 
43Y 


45° 






15 


60°i' 
29*53 


59*2 
30*58 


58" 

32V 


56V 
33*7" 


55°43 

34 '17' 


54*28 
35*32 


53*7 

36*53' 


5t 42 

38*8 


50°i2 
38*48' 


48*35 

41*25 


46*51 
43°9 


45° 






16 


58*23' 
31 V 


57 V 
32V 


56*19 
33*41 


55°u' 

34*49 


53*58' 

36*2' 


52*42' 

37*8 


51*20 

3840 


49*54 

40V 


48*ez' 

4I°38 


46*44 

43*16 


45' 






17 


56V 
33*1 1' 


55*47 

34°i3 


5441' 
35*9 


53*32 

36*28 


52*18 

37*42' 


5l*o* 
39°o" 


49*38 
40*22 


48°u' 

41 43 


46*38 
43*22 


45* 






18 


5S°ii 
34*42 


54*15 

35*45 


53*7 
36*53' 


51V 

38*3' 


50*43 
39*|7 


4a°s» 

40*36 


48°o' 
42*0 


46*33 
43*27 


4-5.° 






19 


53V 
36*9' 


52*46 

37V 


51*38 

38*22 


50*26 
39*34 


49*1 1' 
40*49' 


47*52 

42*8' 


46*28 
43*32 


45° 






20 


s4W 

37 34 


51*20 

3840 


50V 

3948 


48*59 
4!°!' 


47*43 

42V 


46 24 
43*36 


45" 






21 


51%' 

38*56 


49*58 

40Y 


48*48' 
41°*' 


47*36 
42°e4 


46*£o 
43*40 


45* 






22 


49*46 
40° 14 


48*39 
41 V 


47*29 

42*3 r 


46*16 
43*44 


4-5° 






23 


48°30 
4\%6 


4723 

42V 


46°i3' 
4347 


4-5° 






24 


47V 

4243 


46*10 
43°so 


4-5° 






25 


46*7' 
43*53 


45° 






26 


4.5° 







tan a a = _ a 

N 6 



tan a, 



N, 



(See page 13,) 



24 



BROWN & SHARPE MFG. CO. 



TABLE 2. 



Angle of Edge. 

GEAR. 





1 


72 


71 


70 


69 


68 


67 


66 


65 


64 


63 


62 


61 


60 


59 


58 


57 




12 


80*33 

9* £7 


80*16' 
9*95 


80W 

9*46 


80*6 
9*52 


79*58 
IO*V 


79*6t 

10*9' 


79*42 
10*18 


79*32 

lo'ae 


79*23 
10*37 


79° 13 
10*47 


79*3 

10*57 


78*52 
11° 8 


78°4i 
U°I9 


78*30 

11*30 


78j9 

ll°4l' 


78*7 
11*53 




13 


79*46 
10*14 


79*37' 
10*23 


79*29 
I0°3l' 


79*20 
10*40 


79*ll' 
10*49 


79j. 
10*59 


18*51 

11*9' 


78*4 1 
11*19 


78*31 
11*29 


78*20 

ll°40 


78*9 
1 1*51 


77*58 
12*2 


77*46 
12*14 


77*34 
12*26 


77*22 
i2°38 


77*9 
12*51 




!4 


79*6 
II 


78*51 
11*9 


78*41 

11*19 


78*32 
II 28 


78*22 
II 38 


78* « 

11*49 


76* r 
II 59 


77*Sl' 
12*9 


77*40 
12 20 


77*28 
12 32 


77*17 
12 43 


77*5 
12 55 


76*s» 
13 8 


76*39 
13 21 


76°26 
13*34 


76 J I2 

1348 




15 


78*1* 

11*46 


784' 


77*54 
12*6 


77*44 
12*16 


77*34 
12*26 


77*23 
12*37 


77*i2 

12*48 


77*o' 
13*0' 


76*48 
13*12 


76*36 
13*84 


76°24 
13*36 


76*11 
13*49 


7SS8 
14*2 


75*44 
14*16 


75*30 
14 30 


75*15 

14*45 




16 


77*28 

I* * 


77*8 
12*42 


77V 

12*53 


76°S7 
13*3' 


76*45 
13*15 


76*34 
13*26 


76*22 
13*38 


76*10 
13*50 


75*58 
14*2 


75*45 

14*15 


75*32 
14*28 


75*18 
1442 


75*4 
14*56 


74*49 

15*11 


74*35 
15*25 


74*19 
15*41 




17 


76*43 
13*17 


76*32 
13*28' 


76*2 1' 
13*39 


76*10 
13*50 


75*58 
14*2 


75*4* 

14* 14 


75*33 
14*27 


75*81 

14*39 


75*6 
14*52 


74*54 
15*6 


74°4o' 
15*20 


74*25 
15*35 


74*..' 
15*49 


73*56 
16*4 


73*40 
16*20 


73*24 

■ 
16 36 




18 


75*58 
14° i 


75*46 

14° W 


75*35 

I4°2S 


75*23 
14*37 


7S°i6 

14*50 


74°s« 

15*2 


74*45 
15*15 


74°3i' 
15*29 


74°i7 

15*43 


74 ° > 3 
15*58 


73*49 

I6°n' 


"?3°33 
16*27 


73*18 

I6*4z' 


73*2 
16*58 


7245 
17*15 


72*29 
17*31 




19 


7S*ra 
14*47 


75*. " 
14*59 


74*49 
I5*ll' 


74*36 
15*24 


74*23 

15*37 


74* K> 

15*50 


73°56 
16*4 


73*42 

16*18 


73*28 
16*32 


73*13 
16*47 


72*58 

17*2' 


7242 
17*18 


72jzb 
17*34 


it* 

I7°SI 


71*52 
18*8 


71*34 
18*26 




20 


74*29 

15*31 


74*16 

15*44 


74*3' 
15*57 


73*50 

16° 10 


73*37 
16*23 


73*23 

16*42 


73*9' 

16*51 


72*54 
17*6 


72*39 
17*21 


72*23 
17*37 


72S' 
17*53 


7I°SI 
18*9 


71*34 
18*26 


71*16 
18*44 


70*59 

19° r 


70*40 
19*20 




21 


73*45 

Ife'lS 


73*32 
16*28 


73 18 

16*42 


73 ° 4 

16*56 


72*50 
17* to 


72*36 
17°24 


72*21 
17*39 


72*6 
17*54 


71*50 
I8*io 


71*34 
18*2$ 


71*17 

18*43 


71* 0' 
19*0' 


70*43 

I9°I7 


70*24 
19*36 


70V 
19*54 


69*46 
20*24 




22 


73* r 

16*59 


72*47 

17*13 


72*33 
17*27 


72* 19 
17*4 1' 


72*4 
17*56 


71*49 

18* II 


71*34 
18*26 


71*18 
18*42 


71" i 

18*58 


70*45 
19*15 


70*28 
19*34 


70*10 
I9°S6 


69*52 
20*8 


69*33 
20*27 


63*13 
20*47 


6B*54 
21*6 




23 


ltd 

r?*» 


72°3 
I7*$7 


71*48 

I8*u' 


71*34 

18*26 


71'H' 

I8°4l' 


71*3 
18*57 


70*47 
19*13 


70*90 
19*30 


70*14] 

19*46 


69*57 
20 3 


6939 
20°2l' 


69°2(i 
20*40 


69*8 

20°58 


68*42' 
21*18 


68*22 
21*38 


68*2 

21*58 




24 


71*34 
18 tt 


71*19 

18*41' 


71*5' 

18*55 


70*49 
19* II 


70*34 
19*26 


70V 

19*43 


70* r 

19*59 


69*44 
20*16 


69*26 
20*34 


69*9 
20*51 


68*50 
2l°ib 


68°3i' 

21*24 


68*12 
21*48 


67°S2 
22*8 


67*3 1' 
22*29 


67°io 
22*50 




25 


70V 

19*9' 


70*36 
19*24 


70*ti' 
I9°J9 


70 ° 5 

19*55 


69*49 

20*11 


69*32 
20*28 


69*15 
20*4S 


68*57 
■ 

21 3 


68*40 

2 1*20 


68°2i' 
21*39 


68*3* 
21*57 


67*43 
22*17 


67*23 
22*37 


67*2 
22*58 


66*4 1 
23*19 


66*19 
23°4i' 


/. 


26 


70*9 

I9*si' 


69» 
20*V 


69*37 
20*23 


69*2i' 
20 39 


69V 
20*56 


68*46 
21*12 


68*30 
21*30 


68*12 

21*46 


67*54 
22*6 


67*34 
22*26 


67*15 
22*45 


66*55 
23*5 


66*34 

23*26 


66*13 

23*47 


65*51 
24*9 


65*29 
24*3i' 


V- 


27 


69*27 
20*33 


69*10 
20*50 


68*54 

21*6 


68°38 

21*22 


6820 
21*40 


68*3 

21*57 


67*45 

22*15 


67*26' 
22*34 


67*8 
22*52 


66*46 
23*12 


66*28 
23*32 


66*7' 
23*53 


65*46 

24*14 


65*25 
24*35 


65*2 
24*58 


64° 39 
25*2i' 




28 


68*45 
Z 1*15 


68*29 

21*31 


68*12 
21*48 


67*55 
22*5 


67V 

22*23 


67*19 

22*41 


67* 1 
22*S9 


66*42 

23*18 


66*22 

23*38 


66*2 
23*5% 


65*42 
24*18 


66*21 
24*39 


64*59 

25* 1' 


64*37 
25*23 


64°i4 
25*46 


63°so 
26*id 




29 


88*4 
21*56 


6747 

22*13 


67*V> 
22*30 


67*12 

22*48 


66*5* 

23*6 


66*36 
23*24 


66*17 
23*43 


65*57 
24*3 


65*37 
24*23 


65*16 
24*44 


64*55 
25*5 


&4°94i 
25*26 


64*12 

25*48 


63*50 
26*io' 


63*26 
26*34 


63*2 
26*58 




30 


67*» 
22*37 


22*54 


66°48 
23*12 


66*3« 
23*30 


66*12 
23*48 


6S°S2 
24*8 


65*33 
24*27 


65V 

24*46 


G4*63 

25*7 


64*32 
25*28 


64*10 
25*50 


63*49 
26*1 1 


63°26 
26*34 


63*3* 
26*57 


62*39 
27*2J 


62*14 
27*46 




31 


66*42 
23*18 


66*23 

23*35 


66*6 
23*54 


65*46 
24*»2 


65*29 
24*3i' 


65*i6 
24°si 


64°so 
25*io 


64*30 
25°30 


64*9 
25*51 


63*48 
26* re. 


63*26 

26*34 


63*3 

26*57 


62*40 
27*20 


62*8 
27*42 


61*63 

28*7 # 


61*25 
28*31 




32 


86*2' 
23*5^ 


65*44 

24*16 


65*2* 
24*34 


65'V 
24*53 


64*48 

25*2 


64°28 
25*32 


64*8 
25*52 


63*47 

26*13 


63* «b 
26*34 


63f4 
26*56 


62*42 

27*18 


62*19 
27*41 


61*56 

28*4 


61*32 
28*28 


61*7 
28*53 


60*41 

29*19 




33 


65*23 

24*37 


654- 

24*56 


64*45 
25*5 


64*26 
25*14 


64*7' 

25*53 


63*47 
26*13 


63*26 

26°S4 


63*8 
26*55 


62*43 
27*17 


62*21 
27*39 


61*58 
28*2 


61*35 
28*25 


6l*u' 

28*49 


60*47 
29*3 


60*21 
29*39 


5956 

■ 
30 4 




34 


6443 
25*17 


64*25 
25*35 


64V 

25*55 


63*46 

26**4 


6326 
26*34 


6305 

26*55 


62*45 

27*is' 


62*23 

27*37 


62° r 

27*59 


61*38 
23*22 


61*15 
28*45 


60*S2 
29*8 


60*28 
29*32 


60*3 
29*57 


59*37 
30*23 


59*11' 
30*49' 




35 


64° s' 

25*55 


63*45 
26*15 


63*26 
26*34 


63*6' 
26*54 


62*46 

27*14 


62*25 

27* JS 


62°4-' 
27*56 


61*42 

28*18 


61*19 
28*41 


60*57 
29*3 


60*33 
29*27 


60°9 
29*51 


59*45 

30*15 


59*i9 
30*4i' 


58°si 
31*7 


58*27 
31*33 




36 


63*26 
26*34 


63*7 
26°S3 


6247 
27*3 


62*27 

27*33 


62*6 
27*54 


61*45 
28*15 


61*23' 
28*37 


6i" r 

28*59 


60*38 
29*22 


60°i5 

29*45 


59°5i' 
30*9 


59*27 
30*33 


59*2 
30*56 


58°37 
31*23 


58°io 
31*50 


57*13 

32* vi 




37 


62*48 
27*2 


62*28 
27*32 


62*8 
27*S2 


61*48 
28*12' 


61*27 
28*33 


6»°s' 

28°si 


60*44 

29*6 


60*ti 
29*39 


59 St 
30*2 


S9°»s 
30*25 


59*10 
3O°50 


58*46 
31*14 


58 20 
31*40 


57*54 
32*6 


57*28 
32*32 


57° f 
32*59 




38 


62°n" 
27*49 


6l*5l' 
28*9 


61*30 

28*so 


61° 9 
28*5 1' 


6048 
29*12 


60S* 
29*34 


60*4 
29*56 


5341 
30*i9 


59*18 

30*42 


58°S4 
31*6 


S8°*o 
31*30 


58*S 
31*55 


57*39 
32*21 


57°i3 

32*47 


5646 
33*14 


56jt9 

33*4i 




39 


61*33 
28*27 


61*13 
28*47 


60*53 

29*7 


60°3l' 
29*29 


60*10 
29*50 


59*48 

30*12 


59*25 

30*95 


30*5% 


58*39 
31*21 


58° ia 
3146 


5750 
32*10 


57*24 
32*36 


56*S8 
33*2 


56*32 
33*28 


56*6 
33*54 


55*37 
34*23 




40 


60*57 
29*3 


60*56 
29*4' 


60 J ib 
2945' 


59*53 

30° 7 


5932 
30*28 


59*10 
30°SJ 


58*47 
31*13 


S8°» 

31*36 


58* 0' 
32*o' 


57*35 

32*26 


57°id 
32*50 


56*44 

33*16 


56*i9 
33*4i' 


55*52 
34*e 


S5°24 

34 as 


54*57' 
35*3 




41 


60*2* 
2940 


60° 0' 
30*0 


59*39 
30*21 


59*17 
30*43 


58°SS 
31*5 


58*32 
3t <- 28 


58*9 
3t*5l 


57*45 
32*15 


57°2j' 
32*39 


56*57 

33*3 


56*32 
33 2* 


56*6 
33*54 


55*39 

34*21 


55*.Z 
34*46 


54.V 

35*6 


54*6 
35*44 




42 


5945 

30*ii 


59V 
30*36 


S9>' 

30*57 


58'4« 
31**0 


58*16 
3142 


32*5 


57*3i 
32*28 


57*8 
32*52 


5643 
33*7 


56J9 
33*41 


55*53 
34*7 


55°27 

34%i 


55*0' 
35*0 


54*33 
35*27 


54V 
35*55 


53*37 
36*23 



PROVIDENCE, R. I. 



25 



TABLE 2. — {Continued?) 

Angle of Edge, 
gear. 



56 



55 



54 



53 



5251 



50 



49 



48 



47 



46 



45 



44 



43 



42 



12 



77si| 
12*6 



77 42 
i2*ie 



77 28 
12*32 



12*45 



77*o 
13* o' 13 



76*46 



76so 
3*30 



76i* 

1346 



7558 

14*2 



75 41 
14*19 



75*23 
14*37 



75*4 

14*56 



74*46 
15*15 



7435 
IS°3S 



743 
»S*S7 



13 



76°E6 

13V 



2 1 76*2 

13*32 



7613 
13*47 



75*66 

14*2 14 



7^42 



75*26 
14*34 



7S°8 
14*52 



74°5i 
15*9 



7432 
15*28 



74 63 
15*47 



73-53 

leV 



7332 
16*28 



73"i 1' 
16*49 



7246 
17*12 



7£*4i 

17*17 



14 



75*58 
14° 2 



75*43 

14*17 



75K 

14*43 



74°56 74^ 
15*4 IS* 



74*21 
15*39 



74°3' 
15*87 



73%4 

16*16 



73*25 
16*35 



73*4 
16*5*1 



72 21 
17*39 



71*58 

raV 



7134 
18*26 



15 



7SV 
15*o 



7444 

is' 



V 74* 

16 15* 



74 12 
15*40 



73°S5 73* 

16* s' 16* 



37-73 



» 8 

16*42 



72 ss 
17° 1 



72 39 
17*21 



72~I8 
1742 



71*56 

18° * 



71 34 

!8°efe 



7110 

18*50 



70 46 

19° *i 



70*1 
I9*js 



16 



74*3 
15*57 



7347 73 30 



16)3 



16 36 



73 12 

16*46 



72*54 72' 
I7°6' 17 



72.5 
1745 



71 55 
18 5 



7134 
18*26 



71" 12 

ie°4c' 



70 49 
I9*ll' 



»9*34| 



70' 
I9 l 59 



69 35 

20*as 



69V 
20*51 



17 



737 
16*53 



72*49 72*31 



17 11 



17 29 



7213 
1747 



71*54 71 

18*6 18 



18*47 



7052 
19*8 



70» 

19*30 



70*7 
19*53 



6943 
20*17 



69*17 
20*43 



68*52 
21*8 



68 

21*34| 



26 67*58 



222 



18 



72 11 
17*49 



71 



18 7 



S3| 71 
I 



34 



8 26 



7l°is 

18*45 



70*5* 
I9°g' 



7012 
19*48 



69 so 
20*ib 



69*26 



20 34- 20 57 



69 3 



68 38 
21*22 



68 12 
21*48 



6745 

22*15 



67 17 
22*43 



6648 

23*12 



19 



71*15 
18*45 



70 57 

■9° r 



7037 
19*23 



7017 
19*43 



6956 
20V 



69 12 

20*48 



6848 
21*12 



68 25 
21*35 



67^9 
22° 1 



67 34 
22*26 



67 6 
22*54j 



66 38 
23*22 



66 10 
23*50 



65*39 

2**21 



20 



70°2l' 
19*39 



70°. ' 
19*59 



69 41 
20*19 



69 19 
20*4i' 



68 



68*57 
21*3 2 



68ji2 

21*48 



6748 

22*12 



6 V 3 

22*37 



66*57 
23° 3 



6630 
23*30 



66*2 
23*58| 



65*33 65*3 
24*27, 24*$7 



64*32 

25*28 



21 



69 26 69"o 



203* 



20°E 



68*45 
21*15 



6823 
21*37 



68°o' 
22*o' 



67j3 

22*47 



664s 
23*12 



662s 

23*38 



65"SS 
24°S 



65 28 
24*32 



64*59 
25*1 



64w 
25*31 



63'se| 
26*8 



6326 

2eV 



22 



6833 
2f*27| 



67 50 
2»*i| 22*10 



67 2? 
22°3S 



67 4 
22*56 



6640 
23*20 



fefcis 

23*45 



65*49 
24*||' 



6523 
24*37 



64 ss 
25*5 



64 26 
25*34 



63V 
26*3 



63*26 

26*34 



62*54 

27*6 



6221 
27*39 



23 



67V 
22*19 



6ria 

22*42 



66 SS 66'32 
23*5 23*28 



66° 8 
23*52 



6544 

24*16 



6518 
24*42 



64*51 
25*9 



64 
25*361 



24 63 



55 
26* s 



63 26 

26*3* 



62*56 
27*4' 



62*24 
27 



61*52 
e 



36 28 



6f° e i8 
2842 



60*W 
2945 



24 



66*46 

23*12 



6626 



66*2 



23 34 23 58 



65 38 
24*22 



65° |4 
24*4 



64*48 
25*12 



6422 

25*38 



635* 
26*6 



63°2b 
26*3* 



62*57 
27*3 



6227 
27*33 



6I°56 
28*4 



61*23 
28*37 



60 50 
29*10 



£ 



25 



65*57 

24*3 



65 33 

24*27 



65*9 
24*51 



6445 

Z5*i5 



64 to 
25*40 



6353 

26*7 



63 26 

26*34 



feTsil 
27*2 



62^29 

27*3 



61 S9 
28*. 



61*29 
28°3i 



6057 

29*3 



60*24 

29*3« 30" 10 



59 »4 

30*46 



26 



65*6 64*42 
24*94 25*» 



64*18 

25*42 



63*52 
26*8 



6326 
26 r 



62*59 
I 



34 27 



62*3i' 
27*29| 



62 3 

C 1 

2757 



6133 
28*27 



61*3 
28*57 



60 31 
29*29 



5959 
30' 



59*25 

30*si 



58*sd 
3l*io 



58 * 

31*46 



S7*i6 
32' 



2Z, 



64*16 
25*44 



63°5i' 
26*9 



63*26 
26*34 



63°o' 
27V 



62 34 
27*26 



62*j6 
27*5<i 



61 3a 

28*22 



618 

29*52 



60m 
29 f 



60*7 
► t 
S3 



»29 



59b 

30*25 



59* 
30*S8J 



5826 
31*32 



57*53 
32*7 



28 



29 



63*26 
26*34 



[63V 

26*59 



62 36 
27*24 



62*9 
27*si' 



61*42 

28*18 



61*14 

28*46 



60 45 
29*5 



60is 
29*45 



5945 
30*15 



59 13 

30*47 



50*47 

3l e 2»' 



58*7 
3I°53 



62*37 
27*23 



62*12 
27*48 



61*46 
28*iS 



61 19 
2S4i' 



6051 
29*9 



60*23 
29*37 



5953 
30*7 



59 23 

30*37 



58 52 
31*8 



58 19 
31*41 



57*46 
32*14 



57*12 

32*48| 



57*32 
3 2*28 

56*37 

33*23 



S6°56| 
33*4 



S6*i» 
33V 



56*^ 
34° o 



55*23 

34*87 



30 



61*46 

28°i 1 



6rt» 

28*37 



60s? 
29*3 



60*29 

29°3i' 



60° 1 
29 



99 



59 32 S9 2 

30*281 30*56 



58 at 
31*261 



58°o 
32*0 



S7J7 
32*33 



56*53 
33*7 



56*19 
33*4l' 



S5°4i| 
34*17 



55*7 

34*531 



54*28 
35*32 



31 



61* i 

28*53 



6016 

29*a* 



606' 
29*54 



59\i 
30*39 



5944 
30*19 



46 



i 58*42 
31*18 



58*12 
31*48 



5741 
32*19 



57*8 
32* 



56*36 56* » 
"24 33*59 



52 33 



55*26 
34*34 



S4so 
3S*|0 



S4i2 

3S*48| 



53*34 

36*26 



i|57 54 
32' 



32 



CO 15 
29*45 



59*46 



5852 

31*8 



'26 



5723 
32*37 



S6S2, 

33*8 



56i9 
33*41 



55 45 55 i» 



34 IS 



3449 



54 35 

3S* 2 i 



53*s» 
36*2 



53*21 
36* ' 



39 37 



5242 
IB 



33 



5929 
30*31 



59*2|58*34|58°5 
31*55 



30 ss 



3126 



36 



5 57' 
32 



56*34 
33 



56 1 
33*58 



55*3C54° 
34*30 35* 



^56 54 21 
35*3»| 



5345 
36*ib 



53*8 

36*52 



5229 

3731 



5l°so 

o , 

38 9 



34 



5844 
31*16 



58 16 57*48 
32*12 



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S7;i9 

32*4i' 



49 



56 19 
33*41 



5547 
34*13 



55ji5 

34*45 



S4V 
35*19 



54*7 

35*53 



53*32 
36*26 



52S2 
37*8 



52*» 

37*42 



51*40 
38*20 



51* 0' 
39*0' 



35 



58 o 
32*0 



32 57 

> ' * 

29 32 



57 



5633 

33*27 



3 55' 
W34* 



55o 
35*0 



54» 
35*32 



5354 

36*6 



53 20 
36*40 



5244 
37*i6 



52 © 

37*B2 



Sl°30 

38*30 



505i 
39*9 



50*12 
39*46 



36 



5716 



5648 



3244 3912 



'19 



55*49 

34*11' 



55« 



344a 



is 54' 
35' 



54*15 

354sl36 



5342 
18 



53*8 
36*52 



52"33 51 57 



37 27 



38 3 



51*20 
38*40 



50*43 

39*»7 



SOV 
39* 



49^24 

36 



56 40 



37 



5632 
33*28 



56' 



33 



>*4,' S5 
1*56 34' 



55 5 

34*55 



55°7 
34*39 



34 54 

> • I 
26 35 



SS 



53: 
36°: 



52 &fc 

O , 

374 



5223 

37*37 



51*47 

38*13 



51*12 

38*48 



50*35 49*56 49i7 



39zs40 4 



4043 



4837 
4»°23 



38 



5 V'. 
34*9 



54*52 
3S*8 



5423 

35*3; 



5246 
37*14 



S2*tt 
37*46 



38*22 



38 5) 



38*57 



50 27 
39*33 



4949 

40*ll' 



49 11 
40*49 



♦832 

4l*t€ 



47la 
42*8' 



39 



55*9 
34*5 i 



54°39| 54*10 
50 



35*21 



35! 



53 3S 
36*21 



53*7 
36*53 



f;- 



5I°2»| 

38*3 



50 54 
39*6 



50 (9 
39*41 



49' 
40' _ t8 



42 49 



S 
40~S5 



48 27 
41*33 



4746 

42*12 



477 

42si| 



40 



5428 53 £8 



5258 



3532 



36 2 



53*28 
36°32 37*2 



L°26|5I 
*34 



S^SI 20 



5046 SO 12 



38 6 



3840 



3914 



3948 40 



49"36J48*59l48\2T47%iJ47 S 5; 

24 41*1 



46*24 
4 r38|42''t6 |42°S5(43 C 36 



5348 
36*12 



S3 17 
36*43 



52*48 
37*2 



5216 
37*44 



I°4S|S 

> 1 

15 



I 12. 



5039 



38 



5 v 

3848J39'2i 39*55 



30 48 
30 41* 



5446 17 
6 41' 



43 42 



47*40 

20 



47*1 
42* 



4622 

38 



59 43 



45V 
44*19 



42 



53*8' 
36*52 



52*36 
37 



52' 



'22 37' 



51*36 

38*24 



38 



1*4 50*S2J49' 
56 39*28 40' 



58 49 24 4649 
2 40 36 4l*u' 



4813 
41*47 



47*36 46*59 46*20 
42*24|43 l' 43*40 



4^40 

o 
4420 



4.5 



26 



BROWN & SHARPE MFG. CO. 



TABLE 3. 

Angle of Face, 
gear. 



41 



40393837363534 



333231 



30292827 



12 



13 37 
70*34 



13 57 

70 3 3 



W- IB 

70*6' 



14' 39 
69*s4l69* 



15 a* 
6832 



15*9 
6759 



16 15 
67*2 3' 



16 43 

66*4« i 



7*13 
fcfc* s' 



7*3 

65ii 



18 is 
64*« 9 



18 51 
6 3*5 3 



19 27 

63*3' 



20 5' 
62*9' 



13 



14 55 

69*51 



15.17 

69*i« 



15*38 
68V 7 



16! 1 
68°is 



16 a 
674 3 



16*51 
67*9 



17 1 
6633 



174 6 
6556 



18 i» 
65 16 



18 V 

64*34 



19 2 J 
63*5 



19*57 

63*5 



20*32 

62* 



2T11 

6I°23| 



21*54 

60*2 • 



14 



16 w 

6831 



16*34 

66°c 



16 59 

6 7*29 



I 7 U 
66 4-6 



17 50 
6622 



IB 17 
6547 



18 45 

6,S*s 



19 16 
64*30 



19 48 

63*48 



20 to 
63°6 



205 
62*20 



21 s+ 

61*32 



2 2°» 
60°4i 



2*5*1 
59*4 



23.38 

58°5o 



15 



17*28 

67*6 



17*53 

66°45 



1 8°. 8 
66*14 



1 8 44 

65*o 



)9°n 
65*3 



I9 C 40 

64° 84 



6349 



2044] 

6i°6 



ei i8 

62*14| 



21 53 

6i°39 



22*3, 
60*«, 



2 3 ,o| 
60*2 



24*4.2 
5 8*3* 



23*51 
5S'b 



2435| 
58*3 



25*to 
57?4 



16 



18*4 2 

66 V 



I9°9 

6533 



19*35 
64*41 



20°3 

64*28 



63 4 



2l e s 
63% 



2l°3e 

6 2*6 8 



22°9 


6145 



22°*«i 
6l"0i 



23°a 
60*4 



27501 
58 4 2 



24° 1 

59 '2 5I 



57 



2 6°, 2 

5642 



27°i 
55is 



17 



19*56 
64*54 



20 2 4 
64*20 



20° s 
6345 



21*37 

63*9 



21 a. 
639 



2/*43 

62j 



22(4 

6 I 50 



2257 

61*9 



2333 
60 25 



24io 

5 9 40 



25°3, 
58*. 



26 i*J 
57*i 



2 759 
56*13 



27*7 
55° 5 



28 97 

54*ia 



18 



21 9 

6345 



22*6 
6234) 



22*38 
6 1*56 



2o°8 

61*17 



2343 

60*3 



24 ie 

59°5* 



24«« 
£9*8 



25 34 
58 20 



26(5 
57*3i 



26 57 
56*39 



5546 



28 29 

54*49 



29 ie 

53 50 



309 
52 47 



19 



2 2 to 
62 < 3« 



22 49] 

6**, 



23*2 

61 24 



23*52 

60*4 



24V 
60* 



2540 

58*54 



25*1 
59ii 



2537 
5837 



26 1 5 
57*5f 



26*56 
57% 



2738 
56*14 



28*22 

55*2 1 



29°a 
54*24] 



29 56 
53*28 



3043 
52*28 



31*40 
51*2* 



20 



23 30 

61*30 



24 1 

o 
6053 



2432| 


60 iA. 



25*6 

O 

5934) 



26 it 
36°. 



26 55 
5 725 



27e* 

56*38 



28)5 
554» 



28 58 
6 

54 58 



29°44| 

54 



30 s. 
53*9 



3 1 a, 

52*9 



32 iJ 

51*9 



33 c » 

50 *A 



21 



2439 
60°2s 



2S°/o 

59*46 



2 5*3J 
59% 



26 is 
58*26 



26*53 
574S 



27ao| 
57*0 



28*io 
56*/4 



28 50 

6 

552* 



29 St 

54*36 



30 17 
53*8 



31 
52~so 



3/52 
51*32 



32**s 

50*53 



33 36 
49' 



34 31 



22 



25*6 
59 s «o 



26*9 
58*4, 



2653 

3a*i 



2727 
5 7° 9 



28 v « 
56*3« 



28**3 

555 



2922 

554 



30 5 
54*7 



3049 
53°26 



31 34 

S2°3 2J 



32 2 2 

5/*3 



33° (l 
50*4i 



3+°3 

494. 



34 57 

48 3 7 



35*54 

4732 



23 



26 42 

58*6 



2 7°2» 

5738 



28*0 

56*S6 



2836 

56*14 



29i4 
55*30 



2958 
5»*4j 



3535 
5357 



3l°ie 
53*8 



32°i 
S2*,s 



324« 
5/*84 



33°36| 

50*2 



34*i 



8 49*9 



35 20 



36 i5 

4-7*2 7 



37", 2 



37%o 
46*. 8 



24 



2757 
5 7*5 



28 3 
56*35] 



E9°7 



2943 
55*ii 



30*22 

54*2 



3I°2 



3l°45 

52°5i 



32*2a| 
52*2 



5/°i< 



34° 1 
50*5 



34*3 

49V 



42 

49°22 



36*77 

47*,« 



36°3i 
472 



3B°2e 
45 12 



25 



28'ss 
56*i5 



29 34 

a 
5534 



30T 
54*34 



30 c u 

54*42 



30*3 
54*9 



31 29 
53*23 



32° 
52 36 



32*5 2 
51*48 



33*37 
50*47 



34°2i| 

50°5 



34*44 
496 6 



35 "h 

49 °n 



36°0 
4B*6 



37*7 

46*i5 



38°4j| 

4S°! 



3941 

6 

iW* 5 



26 



30 
S5'is\ 



3! 
53 «] 



3 I "54 

53°e 



32°3 
52°2i 



33 is 

5 1*35 



33 58 

50*6 



3531 

6 

49 3 



35 19 

4 8% 



37io 
47*2 



38 u 2 
46°i2 



4s*io 



38 46 

45*0 



39 54 

6 

44 7 



4052 
4-3*2 



27 



3I U 3 
54°,9 



31 39 


53 37 



32 ie 
5 2°«4| 



32 57 

5 2°9 



33 37 
51 23 



34eo 

50 



35°i 
c 

49 37 



35"3 

o 
45 



3*49 



3543 

48*55 



3b 36 
48°2 



3725 

4.7*7 



38", 
46*,o 



41 1 
4 3*5 



42 v 



29 



32*2 

53*22 



32*39 
52*39 



33°i8 

6 
5IS6 



33 57 
51°.. 



34 39 
50*25 



36 v 7 

o 

48 4.7 



36 42 

o 
4746 



3 7*o 
47°2 



38 20 

46 b 



39\, 
451, 



40°|4 



419 
4-3"9 



4-2 7 



29 



30 



31 



32 



33 



34 



35 



36 



37 



38 



39 



40 
41 



32 59 
52*27 



3338 
5 1 44 



34.7 
5l°i 



3458 

50 ,6 



3539 

o 
49 29 



36*23 

|48*4 



37 a 
4750 



3754> 
46* 



38i 



8 46 4, 



39*32 

45 



4024 

It 



4l°ie 
4-3 V i* 



*?> 



3367 

51 as 



3436 

50 50 



35 is 
50° r 



3 556 

49*2 



36 38 


4-834 



372 
47^4 



38' 



36ii 
46*3 



394J] 
45*9 



40" 

44°i- 



32 41 



26 

43*i7 



42i« 



3453 

5041 



35 3 1 36ii 
49*57149° 3 



36°52 
48°* 



35*6 
4950 



555: 

49° 



37 35 



4739 



3820 
4 6 52 



39 5 
46*i 



39« 

< 

45 



40* 

o 
4415 



4I°32 

a 
43 20 



42.23 



37 u 6 
48*; 



37*8 


47j6 



38 3/ 

4 6 49 



39*15 
45*59 



40°l 
. o 
45 9 



4049 
44°! 7 



4138 
43*24 



4228 



36 6 9 
48*49 

373" 



3 7°, 9 
48*i7 



38°o 
47j2 



38*2 

4646 



39*26 
4558 



40°0 
45*e 



4056 4/44 

44*18 43*1*1 



4233 



2 
48*12 



38 11 

47*2 7 



38*53 

46**s 



33*35 

4547 



40^ 
45°8 



4I°4 

44°io 



4149 
43*29 



4237 



38*22 
4724 



39°3 
46°39l45i4 



39*4>0 D 26 
45*a 



4l°io 
44 20 



4 1 "ss 
43*81 



4241 



ACT** 
45°e 



42 

43*34) 



42*5 



40'0 

4552 



40 4o| 
4-5*8 



40 47 
45% 



471 



4424J 



4ltt 
44- i2 



42"5 

43°»7 



42*8 



42°9 
43*3") 



4132 
44*24 



42 14 
43*40 



42*4 



43°,e 

43*2 

43° 2 



42 1 



PROVIDENCE, R. I. 



27 



TABLE 3. — {Continued?) 



Angle oe Face. 

GEAR. 



262524 



23222 



20 



19 



18 



17 



16 



15 



14 



13 



12 



O 



12 



2 131 

bO's 



2Z'.B 

59* o 



23 a 
58*2 



£4*3 
56*4.9 



25*2 
55*3 2 



26*3 
54" T 



52*39 



28*as 

5/*3 



29Vi 

49 V 



3 
K7es 



32*«4. 3426 



45*24 



4.3" 1* 



30 16 

40 io 



38*i7 



13 



22 37 
59*29 



23*26 
58*2 8 



24*vs 
5 7° 2 



25*3 
56' '11 



2b 
54°5i 



2 7°s 
533 6 



28 m- 
|52°o 



29°2 

J0°39 



30We 
4-9°a 



32 V 

47° fc 



333 

4-5 2 2 



35 10 



3o 5S 



4-3°2o4-|°9 



38*o 



14 



24 25 

B 

574. 



25 

5646 



2 6 s 

55 38 



27 5 

54°2 4 



26 4. 
53°8 



29s 



30 20 
SO20 



3i' 33 
48°4 



32 52 

47 °r 



34 '8 

45*2 



35so 

4-32 6 



3728 
41 24 



39 If 



15 



26" 1 
56° 3 



27 a 



11 se 
53°58 



28 48 

•52 44 



30°0 
Si 26 



31%, 
^O^ 



32,9 

48' 



33 36 

4 7'o 



16 



2 7% 2 
54°3 8 



28 W «M 

53°3 



294-3 

52°2i 



30 V* 
51*6 



3l°so 

49*46 



32 

48"2 2 



34-"/ 2 
46*52 



3531 
AS 19 



34ff« 35 23 
45 2.o4333 



3757 

4 |C 39 



3938 



36°.4 
■43' 



36 4 



38°2j 
1.51 



3957 



17 



29 ac 

53*8 



30 a, 
52 °o 



31*2 

50«8 



3228 
49 3 2 



3335 

48 



3447 

46°47 



36 

4 5° 6 



371., 
43°4 3 



384s| 
42° 1 



40,5 



18 



315 

5141 



32 
50 3 2 



J3 4 

4 9° s 



34 a 

48°2 



35,5 
46a, 



36°2a 

4 5 



37js39s 
43°4s42°,i 



401, 



19 



32°36| 
50°, 8 



3336 

49°8 



34 38 



3549 



47«4 46 36 



3653 
4-5°5 



38 6 
4 3. so 



39°24 
42%o 



40*5 



20 



54° s 
48 si 



35°6 

4746 



36*8 

46*321 



37,6 
4-5 ,4 



38 2 6 
43*52 



39 39 
4-2*27 



21 



35 3 , 
47*39 



22 



23 



38,2 
45°, 2 



24 



25 



26 



3632 
46*28 



3 737 
15°, 3 



3844 
4-3 c 56 



39 54 
42*34 



41 e 



36.52 

46°24- 



37SS 

4*°3 



39 u 
43°se| 



40 6 
4-240 



41 19 



39"s 



4-0 20 
4246 



41 28 



39*29 

44° 3 



40 32 
4252 



41 38 



4043 


42 57 



4(46 



4153 



g a = 90° - (a a + P) 
g b = 90 - (a b + /3) 



(See Page ij.) 



TABLE 4 
Angle of Face. — Gear. 



7271 



706968676685 64636261 



60535857 



12 



7°S3' 



8T 

78°5tf 



WT 

78*33 



B-/41 

73*3* 



8°Z/' 
78/3 



8*28 
78'/6 



8 35' 
77° 53 



8*43 
77*47 



5 51 
77°37 



8' 53 

77*25 



9* o 7 

77° I \ 



9'n 

77* r 



9*26 

76°48 



9" 35 
76' 35 



9*45 
76* 23 



9*55' 
76*8 



13 



8°40 
73 12 



8*48 
78 2 



8' 54' 
7752 



9* a 

7742 



9' 9 
77°34 



9 V 8 
77°2e 



9° 26 
77 3 



9°3i 
76 56 



9*/*3 9° 52 
76*45 76*32 



10' I 

76*19 



10* it 
76 7 



Vi'Zi' 
75*53 



t0"3l 

75*33 



10*42 

75*26 



|0*3J 
7S*il' 



14 



9*26 
77*26 



9*34 
77° 16 



9*42 
77*4 



9°. 50 
76 54 



9' S3 
76*43 



10 8 

76*3* 



10" 16 
76* 18 



\0'es 
76 i 



I0*3S"IQ°4S 

75 55 75 41 



10" 54 
75*28 



II* 5 
7515' 



11*16 

75* 



II 27W°33 
7445 74*31 



II' 50 
74 14 



15 



10- 12 

76°4flJ 



10*21 
76*23 



10° 30 

76° 16 



76 6 



|0"A7 

75*55 



10' 57 
75*45 



11*6' 
75*30' 



II" 16 
75 K 



rer* 

75 3 



If 37 

74*49 



iiMYirss 

74° 35 74* 2t 



12' M 

74*7 



12*22 
7i°50 



12*35 
73*35 



12*4$ 
73*18 



16 



!0°59 
175*55 



75 43 



»r e i7 

75 31 



'26 

75 2 6 



11*37 

75 7 



11.46' 
7454 



II 56 

74* 



4*74 



****"**. 

"27 



'2*17 

74 e '3 



12*29 
73*59 



12*40 
73°44 



12 52 
73*28 



13*5 

73*i J 



13* 18 

72° 561 



7240 



13*45 

7223 



17 



44 

75° id 



ir„54 

74 58 



1 2' 4 
74°46 



2*13 
74*33 



1 2° 24 

74*20 



12*34 

74° S 



2*46' 
73°52 



12 Sb 
73 38 



13° 7' 
73*23 



13*21 
73° 9 



13*32 
7252 



13*45 
72'35 



13 59 
7?'2i 



14* If 
72° 3 



Wat 

71*45 



I4'4e 
71*26 



18 



.2*29|l2°40 

742574*12 



12 54 
74* 



/3* 

73*46 



13*12' 

73*32 



|3°23 
73° 1 3 



13*34 

73*4' 



13*47 

7243 



13*59^ 

7233 



14* \i 
72*18 



14*24 
722 



14*36 
7144 



14*52 
71*28 



15*6 
7I'H> 



15*21 
70*51 



15*36 
7054 



19 



13*14 I3'.2i 
73*40 73 27 



13*3$ 

73 14 



13*48 
73* 



14° 

72*46 



14* if 
72°3l' 



14° 24 
72° 16 



14*36 
72° 



14*43 
71*45 



15' 2' 
7/*28 



15*15 

7l'n 



15'so 
70' ' 



54 70 



IV44 

30 



I5*S9 

70 ri 



16' 1 5 

69°59 



16*31 
69°33 



20 



i3;59 |4; ii 

72*57 72*43 



14° 23 
72*29 



I4'34 
7214 



14*46 

72* 



5*4 
7145 



15° II' 
71*29' 



15*25 
7I°I3 



15° 39 
70*56 



15*52 
70*38 



16* 7 

7021 



i6°2i 

70° 3 



1 6" 37 16*53 

69*45 69°25 



|7«8 
69° 6 



I7*» 

68*46 



21 



443 14° 55 
72*13 7I'59 



15' a' 
7/*44 



I5°2J 
7/29 



5*33 

7/'l5 



15*46 
70 56 



15*53 
7041 



16*13 
70*25 



16° 28 

70*8 



I6°42 

69* S 6 



16*58 
69° *■ 



69*n 



I7°28|17°46 
68*54 6V34J6 e i 



18*2 
14 



18° 30 

167*52 



22 



15*2715*40' 
71*29 7l°i4i 



70*59 



i6*6; 

70*44 



16° 20 
70*28 



16 33 

70'n 



16*47 
69*55 



17*2 

69*38' 



17*16 

69*20 



I7°3J 
63* I 



17*49 

68*41 



16*3 
68*23 



18° zo 
68*4' 



16° 37 

67*43 



18° 56 
67°;* 



13* ii 
67*1 



66*9 



23 



16*12 16*24, 
70*46 70° 30 



16° 38 
70' IS 



16* 54' 
69*53 



17*5 

69*43 



17*20 

69*26 



I7°34 
€9° 8 



I7 e 58 

68'sc' 



18*5' 

68*33 



18*20 

68*14 



18*36 
67*54 



18*54 

67' 



34 67 



19* 10 

14 



19*28 

66*52 



19*48 
66*32 



65°/8 



24 



16*55 
70*3 



69*47 



17*22 
69*32. 



17"37 

63* IS 



17*51 

68°59 



18*6 
68*4fi' 



18*21 
68*23 



18*37 
6ft* S 



18* S3 
67*45 



19*9 
67*27 



19*26 

67V 



19*44 
66*46 



21° I 20*19 
66°2s 66*3 



20*33 
65*41 



25 



7*3917*52 



69 21 



69 4 



18*6 

68'48 



18° 21 
68*31 



18*36 
68*14 



18*52 
67*56 



19**7' 

67*37 



13*24; 

67* /* 



19 4* 
67° 



13*57 

66*33 



2T«4 20;3i 

66° 20 65° 58' 



20*51 
65*37 



21*10 
65' 14 



2129 

64°si 



21*50 

64*28 



22*41 
63°3J 



26 



»°2i 18° 36 
68*39168*22 



18*51 
68*5 



19*6' 
67*48 



19*22 
67* JO 



13*37 
67*13 



19° 53 

66*53 



20*10 

66°34 



£0*26 ZQ'tf 
66° 14 65*53 



21*2 21*21 
65*32 65* H' 



2l*4i; 
64*43 



22 

64°26 



22*2$ 
64*2 



27 



19° 3' 
67*57 



19* 19' 
6/39' 



19*34 
67*22 



19*49 
67*5 



20* 6* 

56*46 



20*22' 
66*28 



20 38' 
66*8 



20*56 
6S*48' 



21 13 

65*23 



2T32 
65*8 



21*50 
54*46 



20*17" 
66*41' 



22*10 
54*24 



22*29 
64° ( 



2243 
63*33' 



23*10 
63*14' 



23*31 
6243 



28 



t9'46 
67*16 



20*1 

66*59 



20 = 32 
66*22 



20*56 
66*4 



21*6 

65*44 



21*23" 

65*25 



2l*4f 
65*5' 



22* 

64*44 



22*1* 

54*22 



22*57 
64*1' 



22*56' 
63*38 



23*17 
63*15' 



23*38 
62° 52 



23*69 
62*27 



24' 2! 
62V 



29 



20*27 
66*35 



2043 
66*17 



20*59 
65*59 



21*16 
65*40 



21*33 
65* 2 1 



21' 5C 
65*2 



22 8 
64*42 



22*27 
64*21 



22*45 
63*59 



23*5 
63*37 



23*25 

63*15 



23*44 
62*52 



24*4 
62*28 



24*25 
62*5 



24*48' 
64*44 



25'ia 
64*14 



2T5? 
60'27 



30 



21*3 

65*55 



21*25 
65*37 



21*42' 
65*18 



21*58' 
64*58 



22 15 

64*39 



22=34 
64° 18 



22*52 
63*58 



23' 10 
63*38 



23*3$ 
ai lo 



23*50' 
62*54 



24*io 
62* 3d 



24*30 
62*7 



24*51' 
61*43 



25*12 
61* 18' 



25*36 

60*54 



2?4J 

55*42 



31 



21° 50 

65*14 



22*6 
64*56 



22*24 

64*36 



22*41 

64*17 



22 59 
63*57' 



23*17 
63*37 



23*35 
63*15 



23*SS 

62*55 



24*14' 
52*32 



24*34 
62° 10 



24*54 
6i°46' 



25)7 

61*23 



25*38 

60*56' 



2r5g" 
60° M 



26*22 
645*8' 



32 



2231 
64*35 



2248 23*4 
64°|^ 63*56' 



23 23 
63*37 



23°4C' 
63*16' 



23 59' 
62*55 



24 18 
62*34 



24 3S 
62° I Z 



24*58' 
51*50' 



2ri8' 
61° 26' 



25*33 26° I' 
61* 3' fctfy} 



26*23 2T43 
63° ir> 59*49 



27*9' 
59*21 



2r 3 4 

58°56 



33 



23*io 
-63*56 



2T28 123*46 
63**6lS3*i6 



24*4 
62*54' 



24*22 
62*36 



24*4l' 
62 15 



25*/' 

tei's-i 



2r2i 



25*42 
61*8' 



26*2 

63*44 



26° 24 26j45 
60*20 59° 55 



27*9' 
53*31' 



27*31 
59*5' 



27*56 

58*38 



28*i3 
58°n 



34 



23* Si 
16317 



24*8 
62*5: 



24*27 
62*37 



24'4425 



S216 



4; 

61*56 



25*23 
61*33 



25*42 
61*12 



26° 3' 

50*49 



26°24 
60*26 



26°4* 
60*2 



27*7 
59*37 



d7'29 

59° 13 



27°52 

58*4% 



28 16 
5*822 



28*AO 
57*. 



54 37 



a 



61* IS 



35 



24*29 

62*39 



24°48 

62*11 



25*6' 
61*58 



25*25 

6 1 ' '37 



26*4 
60*54 



26°24 
60*32' 



26=45 
60*9 



27°6 
5944 



27*28 

59*22' 



27*50' 
58*56 



28*13 
58° 31 



28°36 
58*6' 



29* f 

57*39 



29*25 

57*11 



29*56 
5444 



36 



25*9 
62*1 



25*27 
6l*4l' 



25*45 
61*20 



26*24 
60*36 



26°4S 
60*15 



27*5 

59*51 



2r 26 ' 
5928 



27*43 
59*4 



28*io 
58*40 



28°33 

58 \i 



28;S6 
5750 



29°20 
57*24 



29*43 
56° 57 



30*9' 
56°29 



30*33 

56°r 



37 



25*41 
61*" 



26° 6' 
61*2 



26*25 26 
£041 Z'J 



^44 27 °4 
2C 53*58 



27*25 27*45' 
59 35 59° 13 



28'/ 
58° 43 



28'23 
58*25 



28*51 

58° r 



29 IS 

S7 35 



29*38 
57 10 



30*7 
56*42 



30*27 
56 15 



30*52 
55° 46 



31*18' 

55M 



38 



26*25 
60*47 



26*44 
68*26 



27*4 27 

60 4' si 



27*42 i 28' 
59* 23 59* 

58*58 



'24|27 c 44 
42 5920 



28*4 

58*56 



28*26 
55 34 



2847 
58° 9' 



29 q *9' 
57°45 



29*33' 
57°2l' 



29*5 i 
56*55' 



30^20 

56 30 



30*44 
56*2 



31*9 

55*35 



31*35 
55*7' 



32*1 

5433 



39 



27*5' 
60° 9 



27 e *22 

59*48 



28*22 

53*42' 



28*43 

1 58* 1 3 



29*5' 
5755 



29*27 
57*31 



29*49 
57° 7 



30*13 
56*4i 



30*36 
56° 16" 



31*1 

55*49 



31*26 
5522 



34*50 
54 54 



32*16' 
54 



3243 
57 



28 53 



40 



pr4o 

59*34 



2r40 
59*32 



29 I' 

58*5 



29*22 
57*42 



29*43 
57*17 



30*6 
56*54 



30/28 
56 28 



30*52 
56 2' 



31° 17 

55*37 



31*42 
55*i6 



32*6' 
54*44 



32 . 3 ! 
54*15 



32*57 
S3°4€ 



3324 

53*18 



41 



2rir 

58*57 



2*Tj7 
58*37 



28*S7 
5*8* is 



57 39 57° 



29°39' 
57*29' 



30*1' 
57 4' 



30 5 22 

56*40 



30*45 
56*15 



3i; 9' 
35* 5\ 



3l*5i 
55* 25 



31*55 

54*59 



32*20 
5432 



3246 
54° 4 



33*12 

S3* 36 



3V 39 
53*7 



34 6 

52 38 



42 



26' ii 
58*22 



29*t£ 
58* t 



30*16 
56*52 



30*38 
56*28 



31 

56*4' 



31*23 
55' 39 



31*47 
55* 13' 



32' 10 
54*48' 



32*35 

54*21 



33* 
53*54 



33*26 
53*26 



33* si 
52 58 



34 »9 
52*23 



34'46 
52° 



TABLE 4. — (Continued?) 
Angle of Face. — Gear. 



29 



565554 



53 



52 



51 



5049484746454443 



42 



10° 6' 
7554 



10; »6 

7540 



»*.28 
7524 



10° 39' 
75° 9' 



10*52 
7452 



11*3' 
7437 



11° '5" 

74 is 



If 30 
73*58 



11*43' 
73*39 



IIM 

73° 20 



12* 13 

72 53 



72 37 



12*45* 
72° 1 5' 



13*1' 
71° 51 



I3*t9 
71*25 



3 



11*4' 

74 56 



II" 16 

7443 



ir o 2s 

742* 



11*4-2 
74 8 



11*54 

73° 50 



12 8 
73*32 



12* ao 
73° 12 



|2°37 
7253 



12*51 

72 33 



13*7 
72° II 



13*23' 
71*49 



1340 
7l°26 



13*58* 
71° 2' 



14*16* 
70° 38 



14*35 
70°<i 



12* 2' 
73*58 



12*16 
7342 



12*29 
73*25 



12*43 
73 7 



12 57 
72 4Q 



15 II" 

72*29 



»3*26 
72 8 



1S4I 
71° 4B 



C3°53 
71*27 



14° IS 
71° 5 



70°4I 



14*51 
10° 17 



15° 10 
59 52 



15*30 



15 51' 



69 26 68 59' 



13° I 
73° I' 



13*16 

72*44 



13*28 
72 26 



13*43' 
72 7 



13*59 
7!*49' 



14*14 
if 28 



14*3© 
71° 6* 



14*47 
70*45 



15° 5' 
70*23 



15*23 

69*53 



iS Q 42 
69*34 



16° 



16° 22 
6842 



16*43 
6815 



17*5* 

67*47 



13° 59 
72*5' 



14*13 
71*47 



14*28 
7I°28' 



14*44' 
71° 8' 



15; r 

70*39 



15*17 

7027 



sras 
70° 5 



15*52 
69*42 



«6>* 

69*19 



16° 3d 

54 



16*56 
68° 28' 



17*10' 

68*2* 



17*32 

67*34 



17*56 
67*6' 



18* 18' 
66°36 



7 



14*57 
71*9' 



I5*n* 
70*49 



«5* sa" 
70*30 



5*44 
7010 



16° r 

69*49 



16*18 

69*26 



16° 37 
69° 3' 



16*56 

68*39 



17*15 

S3* 15 



17*36 

67°s6 



17*57 

6723' 



58*20 
66*S4 



18*43 
66°27 



K 6 ' 

65*58 



19*31 
65*27 



is; 52 
7014 



16*7' 

69*53 



16* 26 
69*34 



16*42 
63*12 



17*1' 

68*43 



»7°2o 
68*26 



17*39' 

68*3 



17*58 

67*38 



18*20 
67° 12 



18*44 
66*47 



19* 3' 
66° 19 



1927 
65*5l' 



19*50 
65°2o 



20*18 
64°5fl 



2042 

64 98 



9 



16*49 
69*19 



17*2 

68*58 



•7.23 
6837 



17*41 
68*15 



18° 

67*52 



18*21 
67*29 



18*40 
67*4 



19*1 
66*37 



19*22 

66*12 



19.46 

65*44 



20° 8' 
65° 16 



20-. 
64*4 



'34 20' 



59 
6415 



2»°24 

63*44 



21*52 
63*|0 



20 



17X4 

68*26 



'1 18* 19' 

°3* 67*44' 



18*40' 
67*18 



66°S4 



19*20 

66*36 



1941' 
66*5 



20*2' 
65*38 



20*25 
65*1 1' 



20*45 
S443 



21* 13' 
64*13 



21*39 
63*43 



22 5 
63*11 



22*32 
62*38 



23" 

62*4* 



21 



18*39 
61° 31' 



18*57 
679' 



19* 16' 
66 s 



20*1? 

65*52 



4666 



19*37 

23 



19*58 
65*58 



20J9 
65 33 



20*41 
85*7 



21*3' 
64° 



21*27 
II 



39 64 



21*52 

63*42 



22*17 
63*13' 



22*43 
6241 



23°'o' 
62° 8' 



23*38 
61*34 



24*8' 
Sl° 



22 



19*32 
66 38 



19*52 
66 16 



20 33 
65*27 



20*55 
65° 3* 



2117 
6437 



21*40 
6410 



22° 3' 

63V 



22*0 

6313 



22*53 23*19 
6243 62*11 



23*46 
61*40 



24* l* 
61*7' 



24*44 
60*32 



l*fi4 

59 56 



23 



20*25 
6547 



20*47 
65*23 



2f8 



21*29' 



6458 6433 



21*52' 
64 8' 



Zt 13 
63*41 



22*37 
6313 



23*2* 
62*44 



23 27 
62*15 



23*54 
61*44 



24*21 
61° 13 



24*49 
6044 



25*18 
60*6' 



25*47 
53*31' 



26*18 
58*54 



24 



21* »9' 
64*55 



21*39 
64*31 



22*1' 

64*5' 



22J24 
63*40 



46 23 



22 
63*14 



|0 

6246 



£3*36 
62* 19 



24' 
61*48 



24°26 
61° 18 



24*53 
60°47 



25*21 25*49' 
60*15 5941 



26*26 
59 6' 



26*51 
58*31 



2723 
57*53 



25 



22*11 
64° 5' 



2233 

6339' 



22*56 
63*14 



23*18' 
62*48 



23V 
62*2* 



24*7 
61*53' 



24 ° 

6l°24 



3i24 



57 

SO" S3 



25*24 
60*22 



25*52 
59*50 



26*20 
59* 18 



26*50 
5844 



27*21 
58*3* 



27°tt 
57*32 



274? 28*11 
5747 3/ II 



28*26 
56 54 



29*27 
55*5S 



o 
rz; 

P4 



26 



23*3' 
63°»5 



23*25 
62*49 



23*47 
62*23 



24°|3 
61*56 



24*25 
61*28 



25*1' 
60*59 



25*28* 

6030 



25*S3 
53 59 



26*ai 
59*27 



2T49 
53*55 



27°I9' 
58*21 



28*J4 
56*34 



27 



23*53' 

62*25 



24*16 
£1*58 



24*40 25' 



6132 



*5* 

6!° 5' 



25*29 
37 



25^? 
6043 



25*55 
60*7 



26*22 
59° 38 



26*48' 
59*5 



27*17 
58*33 



27*46 



21*41 
57*8' 



28*16' 
57*26 



28*47 
56*51* 



29*19 
56*15 



29*52 
55*38 



30*2? 
5459 



28 



24*44 
61*36 



25*7 
61*9 



25*56 
60 14 



25*22 
59°46 



26*48' 
59*16 



27* J5 

58*45 



27*43' 
58* 13 



28*12' 
57*42 



29*12 
56*32 



29*43 
&1 



30*16' 

55*2<i 



30%tf 
54*42 



3T25 7 
54* 3* 



23 



25*33' 
60*47 



25*57 
60*21 



26*22 
59*52 



26*47 
59*25 



27*14 

58*56 



27*40 
58*26 



28*8 
57*54 



28*36* 
57*22 



29*5' 
5649 



29*37 
56*15 



30*8' 
55*40 



30*40 
55*4' 



3**I3 

54*27 



31*48 
53*4%' 



3221 

53*9' 



30 



26* s* 
60' 



26*47 
5933 



59*6* 



27*38 
58*36 



28*4 
58*6' 



26.3Z 
57*36 



29* 
5/4 



29*28 
56*32 



29*58 
55*38 



30*36 
55*34 



31*2* 
5448 



31*34 
5412 



32*8" 
53°34 



32*44 
[52 54 



33*19 

52* I5" 



31 



27*10 
59* «4 



27*34 

5o°46' 



28*3' 
58*15' 



28*27 
57*4S' 



28*54 
57 18 



29*23 
56*47 



29*33'* 
56 4l' 



29*51' 
56" 15 



3ft* 26 
S5*42 



30*52 
55*8* 



31"22 

5434 



3I°SS 
53*57 



3229 
53*21 



33*2 

52°42 



33*39 

52*3' 



34*15 
51*23 



32 



27*56 
58*28 



28*45 
57*43 



28*23 
57*59 



23*43 23*17 
57*31 57* I' 



56* 



3©°42 
55*29 



31° 10 
54*54 



31*42 

54*20 



32*14 

5344 



32*46 
53 8' 



33*21 
52*31' 



33*56 
51*52 



34*3i 
51° 13 



35*8' 
50*32 



33 



29°<0 
57*14 



23*37 

56*45 



30*5 
15 



3032 
55*49' 



31*1' 
55*13' 



31*31 
54*39 



32*1' 
54*5 



32*32 
53*32 



*a*4; 

58*56 



33J38' 
52 20 



34*12 
51*42 



33*54 34*28' 
52 8' Sl°3£ 



34*47 
51*3' 



35*z4 

50 22 



36° 
49*41 



34 



29*31 
56*59 



29*57 
56*29 



30*24 
S6_ 



3«JV 
55*29 



31*20 
54*58' 



31*43 
54*27 



32JI9' 
53*53 



32*50 
53*20 



33*22 

5244 



35° 6' 
50*50 



35*38 
50 14 



36*15 
49*35 



36*53* 

48°S3 



35 



30* is; 

56° 15 



30*42 
55*46 



31* 

55 16' 



10 31 



.** 

5444 



3-rr 

54*13' 



S34C 



33°7' 
53°7 



33*38 
5234 



34*14 
51*58 



34*42 
51*22 



35*n 

5045 



35° 51 
50*7 



36*27 

143 27 



37"S 

48*47' 



37*42 
48 6' 



36 



31 
55*32 



31*27 
55*3 



31*55 
5433 



32 23 
54*1" 



32*53 
53° 28 



33*2^ 
52° 57 



33*53 
52*23 



34 25 
Sl o 40 



34*57 
51° 13 



35*31 
50*37 



36*5' 

49 S9 



49 21 



37 16 

148*42 



37*53 

48*1* 



38*32* 
47*20 



37 



31*45" 
5449 



32*l2 
54*20 



32*40 
5350 



33*8 
53*18 



3338 
52*46 



34*9' 
5213 



34>' 
31*40 



35* 
51°^ 



12 35 



43 
50*29 



36*13 
4952 



36"Sl 
49 15 



37 o** 27 J 38 °* V 
48°37 47*56 



3842 
47*16 



39*26 
4631 



38 



32*27 

S4 9' 



3Ys? 

53 38 



33*24 
53° 8 



3352 
52*38 



34*22 
52 4' 



34°54 
51*30 



35*24 
5056 



35*51 
50 21 



36*29 
4945 



37*3 

43 9 



37*38 
48*32 



38*14 
47*52 



38*51 
41 13 



3928' 
46°32 



40*7' 
45*5l' 



40sT 
45*7 



39 



33° 10" 
53*28 



33 
52 57 



39 34' 



7 
52 27 



34*36 
51*54 



35*7 
51*2 1* 



35*37 
50*49 



36*9' 
50*15 



36*41 
49*39 



37*15 
4-9*3' 



37*48 
48° 



26 47 



38*24 
48 



39 

47*io 



39*36 
46*30 



40*i3 

4545 



40 



33*52' 
52*48' 



34*21' 
52*17 



§4?3? 
52*9 



34*50 
51*46 



35*18' 
5M4 



35*49 

S04I 



36*20 
50 8' 



36°S3* 
49 33' 



37*25 
4€st 



37*58 

48*22 



38*33 

47*45 



39*8 
4/6' 



3944 
4628 



40*20 
45*48 



40*58 
43*8' 



41*37 
44*25 



41 



35*3' 
51*37" 



35°3i 
SlV 



36*1 
50 33 



36*31 
SO* 1 



37*3' 
49*27 



37*35' 

48*53 



38*7' 
48*17 



38*41 
47*41 



39*16' 
47*4 



39*51' 
46 25 



40*27 
45*47 



41° 5' 
45*7' 



41*42 
44*26 



42*22* 
43*44 



42 



35*14 



35*43' 
SO* 59 



36* 12* 
50*28 



36*42 

49S4J 



37*13' 
49* 2l' 



37*44' 
48*48 



38*17 
48*13 



38*49 
47°37' 



39*23 
47* r 



39 58 
46*24 



40 34 
45*46 



41*9 
45*7. 



4147 
4427 



42 26 
43*46 



43*4 



30 



BROWN & SHARPE MFG. CO. 



NATURAL SIXE. 



Deg. 


0' 


10' 


20' 


30' 


40' 


50' 


1 

60' 







.00000 


.00291 


.00581 


. 00872 


.01163 


. 01454 


.01745 


89 


1 


.01745 


.02036 


. 02326 


.02617 


.02908 


.03199 


.03489 


88 ! 


2 


.03489 


.03780 


.04071 


.04361 


.04652 


.04943 


.05233 


87 


3 


.05233 


.05524 


.05814 


.06104 


.06395 


.06685 


.06975 


86 


4 


.06975 


.07265 


. 07555 


.07845 


.08135 


.08425 


.08715 


85 


5 


.08715 


.09005 


.09295 


.09584 


.09874 


.10163 


.10452 


84 


C 


.10452 


.10742 


.11031 


.11320 


.11609 


.11898 


.12186 


83 


7 


.12180 


.12475 


.12764 


.13052 


. 13341 


. 13629 


.13917 


82 i 


8 


.13917 


.14205 


. 14493 


. 14780 


. 15068 


.15356 


. 15643 


81 


9 


. 15643 


. 15930 


.16217 


. 16504 


.16791 


.17078 


.17364 


80 


10 


.17364 


.17651 


.17937 


.13223 


.18509 


.18795 


.19080 


79 


11 


.19080 


.19366 


.19651 


.19936 


.20221 


. 20500 


.20791 ' 


78 


12 


.20791 


.21075 


.21359 


.21644 


.21927 


.22211 


.22495 


77 


13 


. 22495 


.22778 


.23061 


.23344 


.23627 


.23909 


.24192 


76 


14 


.24192 


. 24474 


.24756 


.25038 


.25319 


.25600 


.25881 


75 


15 


.25881 


.26162 


.26443 


.26723 


.27004 


.27284 


.27563 


74 


16 


.27563 


.27843 


.28122 


.28401 


.28680 


.28958 


.29237 


73 


17 


.29237 


.29515 


.29793 


.30070 


.30347 


.30624 


.30901 


72 


18 


.30901 


.31178 


.31454 


.31730 


.32000 


.32281 


.32556 


71 


19 


.32556 


.3283L 


.33100 


.33380 


.33654 


.33928 


.34202 


70 


20 


.34202 


.34475 


.34748 


.35020 


.35293 


.35565 


.35836 


69 


21 


.35836 


.36108 


.36379 


.36650 


.36920 


.37190 


.37460 


68 


22 


.37460 


.37730 


.37999 


.38268 


. 38536 


.38805 


.39073 | 


67 


23 


.39073 


.39340 


.39607 


.39874 


.40141 


.40407 


.40673 I 


66 


24 


.40673 


.40989 


.41204 


.41469 


.41733 


.41998 


.42261 ! 


65 


25 


.42261 


. 42525 


.42788 


.43051 


.43313 


.43575 


.43837 ! 


64 


26 


.43837 


.44098 


.44359 


.44619 


.44879 


.45139 


.45399 


63 


27 


.45399 


.45658 


.45916 


.46174 


.46432 


.46690 


.46947 


62 


28 


.46947 


.47203 


.47460 


.47715 


.47971 


.48226 


.48481 


61 


29 


.48481 


.48735 


.43989 


.49242 


.49495 


. 49747 


.50000 


60 


30 


.50000 


. 50251 


.50503 


.50753 


.51004 


.51254 


.51503 


59 


31 


.51503 


.51752 


. 52001 


.52249 


. 52497 


. 52745 


.52991 


58 


32 


.52991 


. 53238 


. 53484 


.53730 


. 53975 


.54219 


.54463 


57 


33 


. 54463 


. 54707 


.54950 


.55193 


.55436 


.55677 


.55919 


56 


34 


.55919 


.56160 


.56400 


.56640 


.56880 


.57119 


. 57357 


55 


35 


.57357 


. 57595 


. 57833 


.58070 


.58306 


. 58542 


.58778 


54 


36 


.58778 


.59013 


.59243 


. 59482 


.59715 


.59948 


.60181 


53 


37 


.60181 


.60413 


.60645 


.60876 


.61106 


.61336 


.61566 


52 


38 


.61560 


.61795 


. 62023 


.62251 


.62478 


. 62705 


.62932 


51 


39 


.62932 


.63157 


. 63383 


.63607 


.63832 


.64055 


.64278 


50 


40 


.64278 


.64501 


.64723 


.64944 


.65165 


.65386 


.65605 


49 


41 


.65605 


.05825 


.66043 


.66262 


. 66479 


.66696 


.66913 


48 


1 42 


.66913 


.67128 


. 67344 


.67559 


.67773 


.67986 


.68199 


47 


43 


.68199 


.68412 


.68624 


. 68835 


.69046 


.69256 


.69465 


46 


44 


. 69465 


.69674 


.69883 


. 70090 


. 70298 


. 70504 


. 70710 


45 




60' 


50' 


40' 


30' 


20' 


10' 


C 


Deg. 



NATURAL COSINE. 



PROVIDENCE, R. I. 



31 



NATURAL SINE. 



Beg. 


0' 


10' 


1 

20' 


| 

30' 


40' 


50' 


60' 




45 


.70710 


.70916 


.71120 


.71325 


. 71528 


.71731 


.71934 


44 


46 


.71934 


.72135 


.72336 


.72537 


. 72737 


. 72936 


.73135 


43 


47 


. 73135 


. 73333 


. 73530 


.73727 


. 73923 


.74119 


.74314 


42 


48 


.74314 


.74508 


. 74702 


. 74895 


.75088 


.75279 


.75471 


41 


49 


.75471 


. 75661 


. 75851 


. 76040 


.76229 


.76417 


.76604 


40 


50 


.76004 


. 76791 


. 76977 


.77102 


. 77347 


. 77531 


.77714 


39 


51 


.77714 


.77897 


. 78079 


. 78260 


. 78441 


. 78621 


.78801 


38 


53 


.78801 


.78979 


.79157 


.79335 


.79512 


.79688 


.79863 


37 


53 


.79863 


.80038 


.80212 


.80385 


.80558 


.80730 


.80901 


36 


54 


.80901 


.81072 


.81242 


.81411 


.81580 


.81748 


.81915 


35 


55 


.81915 


.82081 


.82247 


.82412 


.82577 


.82740 


.82903 


34 


56 


.82903 


.83066 


.83227 


.83388 


.83548 


.83708 


.83867 


33 


57 


.83867 


.84025 


.84182 


.84339 


.84495 


.84650 


.84804 


32 


58 


.84804 


.84958 


.85111 


.85264 


.85415 


.85566 


.85716 


31 


59 


.85716 


.85866 


.86014 


.86162 


.86310 


.86456 


. 86602 


30 


60 


.86602 


.86747 


.86892 


.87035 


.87178 


.87320 


.87462 


29 


61 


.87462 


.87602 


. 87742 


.87881 


.88020 


.88157 


.88294 


28 


62 


.88294 


.88430 


.88566 


.88701 


. 88835 


. 88968 


.89100 


27 


63 


.89100 


.89232 


.89363 


.89493 


.89622 


.89751 


.89879 


26 


64 


.89879 


.90006 


.90132 


.90258 


.90383 


.90507 


.90630 


25 


65 


.90630 


.90753 


.90875 


. 90996 


.91116 


.91235 


.91354 


24 


m 


.91354 


.91472 


.91589 


.91706 


.91821 


.91936 


.92050 


23 


67 


.92050 


.92163 


.92276 


. 92388 


. 92498 


.92609 


.92718 


22 


68 


.92718 


.92827 


.92934 


.93041 


.93148 


.93253 


. 93358 


21 


69 


.93358 


.93461 


.93565 


.93667 


.93768 


.93869 


.93969 


20 


70 


.93969 


.94068 


.94166 


.94264 


.94360 


.94456 


.94551 


19 


71 


.94551 


.94646 


.94739 


.94832 


.94924 


.95015 


.95105 


18 


72 


.95105 


.95195 


.95283 


.95371 


. 95458 


.95545 


.95630 


17 


73 


.95630 


.95715 


.95799 


.95882 


.95964 


.96045 


.96126 


16 


74 


.96126 


.96205 


. 96284 


.96363 


.96440 


.96516 


.96592 


15 


75 


.96592 


.96667 


.96741 


.96814 


.96887 


.96958 


.97029 


14 


76 


.97029 


.97099 


.97168 


.97237 


.97304 


.97371 


.97437 


13 


77. 


.97437 


.97502 


.97566 


.9762!) 


.97692 


.97753 


.97814 


12 


78 


.97814 


.97874 


.97934 


.97992 


.98050 


.98106 


.98162 


11 


79 


.98162 


.98217 


.98272 


.98325 


.98378 


.98429 


.98480 


10 


80 


.98480 


.98530 


.98580 


. 98628 


.98676 


. 98722 


.98768 


9 


81 


.98768 


.98813 


.98858 


. 98901 


. 98944 


. 98985 


.99026 


8 


82 


.99026 


.99066 


.09106 


.99144 


.99182 


.99218 


.99254 


7 


83 


.99254 


.99289 


.99323 


.99357 


.99389 


.99421 


.99452 


6 


84 


.99452 


.99482 


.99511 


.99539 


.99567 


.99593 


.99619 


5 


85 


.99619 


.99644 


.99668 


.99691 


.99714 


.99735 


.99756 


4 


86 


.99756 


.99770 


.99795 


99813 


.99830 


.99847 


.99863 


3 


87 


.99863 


.99877 


.99891 


.99904 


.99917 


. 99928 


. 99939 


2 


88 


.99939 


.99948 


.99957 


.99965 


.99972 


.99979 


.99984 


1 


89 


.99984 


.99989 


.99993 


.99996 


.99998 


. 99999 


1.0000 





" 


60' 


50' 


40' 


30' 


20' 


10' 


0- | 


I>eg. 



NATURAL COSINE. 



32 



BRUWX oc SHARPE -MFG. CO. 



NATURAL TANGENT. 



Deg. 


0' 


10' 


CO' 


SO' 


40' 


50' 


60' 







.00000 


.00290 


.00581 


.00872 


.01163 


.01454 


.01745 


89 


1 


.01745 


.02036 


.02327 


.02618 


.02909 


.03200 


.03492 


88 


2 


.03492 


.03783 


.04074 


.04366 


.04657 


.04949 


.05240 


87 


3 


.05240 


.05532 


.05824 


.06116 


.06408 


.06700 


.06992 


86 


4 


.06992 


.07285 


.07577 


.07870 


.08162 


.08455 


.08748 


85 


5 


.08748 


.09042 


.09335 


.09628 


.09922 


. 10216 


.10510 


84 


. 6 


.10510 


.10804 


.11099 


.11393 


.11688 


.11983 


.12278 


83 


7 


.12278 


.12573 


.12869 


.13165 


.13461 


. 13757 


.14054 


; 82 


8 


.14054 


.14350 


. 14647 


. 14945 


.15242 


.15540 


.15838 


81 


9 


.15838 


.16186 


. 16435 


.16734 


.17033 


.17332 


.17632 


80 


10 


.17632 


.17932 


.18233 


.18533 


.18834 


.19136 


.19438 


79 


11 


.19438 


. 19740 


.20042 


.20345 


.20648 


.20951 


.21255 


78 


12 


.21255 


.21559 


.21864 


.22169 


.22474 


.22780 


. 23086 


77 


13 


.23086 


.23393 


.23700 


.24007 


.24315 


.24624 


.24932 


76 


14 


.24932 


.25242 


.25551 


.25861 


.26172 


.26483 


.26794 


75 


15 


.26794 


.27106 


.27419 


.27732 


.28046 


.28360 


.28674 


. 74 


16 


.28674 


.28989 


.29305 


.29621 


.29938 


. 30255 


.30573 


73 


17 


.30573 


.30891 


.31210 


.31529 


.31850 


.32170 


.32492 


72 


18 


.32492 


.32813 


.33136 


.33459 


.33783 


.34107 


.34432 


71 


19 


.34432 


.34758 


.35084 


.35411 


.35739 


.36067 


.36397 


70 


20 


.36397 


.36726 


.37057 


.37388 


.37720 


.38053 


.38386 


69 


21 


.38386 


.38720 


.39055 


.39391 


.39727 


.40064 


.40402 


; 68 


22 


.40402 


.40741 


.41080 


.41421 


.41762 


.42104 


.42447 


67 


23 


.42447 


.42791 


.43135 


.43481 


.43827 


.44174 


.44522 


66 


24 


.44522 


.44871 


.45221 


.45572 


.45924 


.46277 


.46630 


65 


25 


.46630 


.46985 


.47341 


.47697 


.48055 


.48413 


.48773 


64 


26 


.48773 


.49133 


.49495 


.49858 


.50221 


.50586 


.50952 


63 


27 


.50952 


.51319 


.51687 


.52056 


.52427 


.52798 


.53170 


62 


28 


.53170 


.53544 


.53919 


.54295 


. 54672 


.55051 


.55430 


61 


29 


.55430 


.55811 


.56193 


.56577 


.56961 


.57347 


.57735 


60 


30 


. 57705 


.58123 


.58513 


.58904 


.59297 


.59690 


.60086 


1 59 


31 


.60086 


.60482 


.60880 


.61280 


.61680 


.62083 


.62486 


1 58 


32 


.62486 


. 62892 


.63298 


.63707 


.64116 


.64528 


.64940 


1 57 


33 , 


.64940 


. 65355 


.65771 


.66188 


.66607 


.67028 


.67450 


; 56 


34 


. 67450 


.67874 


.68300 


.68728 


.69157 


.69588 


.70020 


55 


35 


.70020 


.70455 


.70891 


.71329 


.71769 


.72210 


.72654 


54 


36 


.72654 


. 73099 


. 73546 


.73996 


.74447 


.74900 


. 75355 


53 


37 


. 75355 


. 75812 


.76271 


.76732 


.77195 


.77661 


.78128 


52 


38 


.78128 


.78598 


.79069 


.79543 


.80019 


.80497 


80978 


51 


39 


.80978 


.81461 


.81946 


.82433 


.82923 


.83415 


.83910 


50 


40 


.83910 


.84400 


.84900 


.85408 


.85912 


.86419 


. 86928 


1 49 


41 


.86928 


.87440 


. 87955 


.88472 


.88992 


.89515 


.90040 


I 48 


42 


.90040 


■ .90568 


.91099 


.91633 


.92169 


.92709 


.93251 


47 


43 


.93251 


.93796 


.94345 


.94896 


. 95450 


.96008 


.96568 


| 46 


44 


.96568 


.97132 


; .97699 


.98269 


.98843 


.99419 


1.0000 


45 




GO' 

1 


50' 


40' 


30' 


20' 


10' 


0' 


: Deg. 



NATURAL COTANGENT. 



PROVIDENCE, R. I. 



33 



NATURAL TANGENT. 



Deg. 


o' 


10' 


20' 


30' 


40' 


50' 


60 




45 


1.0000 


1.0058 


1.0117 


1.0176 


1.0235 


1.0295 


1.0355 


44 


46 


1.0355 


1.0415 


1.0476 


1.0537 


1 . 0599 


1.0661 


1.0723 


43 


47 


1.0723 


1.0786 


1.0849 


1.0913 


1.0977 


1 . 1041 


1.1106 


42 


48 


1.1106 


1.1171 


1.1236 


1.1302 


1.1369 


1.1436 


1.1503 


41 


49 


1.1503 


1.1571 


1 . 1639 


1.1708 


1.1777 


1.1847 


1.1917 


40 


50 


1.1917 


1.1988 


1.2059 


1.2131 


1.2203 


1.2275 


1 2349 


39 


51 


1.2349 


1.2422 


1.2496 


1.2571 


1.2647 


1.2723 


1.2799 


38 


52 


1.2799 


1.2876 


1.2954 


1.3032 


1.3111 


1.3190 


1.3270 


37 


53 


1.3270 


1.3351 


1.3432 


1.3514 


1.3596 


1.3680 


1.3763 


36 


54 


1.3763 


1.3848 


1.3933 


1.4019 


1.4106 


1.4193 


1.4281 


35 


55 


1.4281 


1.4370 


1.4459 


1.4550 


1.4641 


1.4733 


1.4825 


34 


56 


1.4825 


1.4919 


1.5013 


1.5108 


1.5204 


1.5301 


1.5398 


33 


57 


1.5398 


1.5497 


1.5596 


1.5696 


1.5798 


1.5900 


1.6003 


32 


58 


1.6003 


1.6107 


1.6212 


1.6318 


1.6425 


1.6533 


1.6642 


31 


59 


1.6642 


1.6753 


1.6864 


1.6976 


1.7090 


1.7204 


1.7320 


30 


60 


1 . 7320 


1.7437 


1.7555 


1 . 7674 


1.7795 


1.7917 


1.8040 


29 


61 


1.8040 


1.8164 


1.8290 


1.8417 


1.8546 


1.8676 


1.8807 


28 


62 


1.8807 


1.8940 


1.9074 


1.9209 


1.9347 


1.9485 


1.9626 


27 


63 


1.9626 


1.9768 


1.9911 


2.0056 


2.0203 


2.0352 


2.0503 


26 


64 


2.0503 


2.0655 


2.0809 


2.0965 


2.1123 


2.1283 


2.1445 


25 


65 


2.1445 


2.1609 


2.1774 


2.1943 


2.2113 


2.2285 


2.2460 


24 


66 


2.2460 


2.2637 


2.2816 


2.2998 


2.3182 


2.3369 


2.3558 


23 


67 


2.3558 


2.3750 


2.3944 


2.4142 


2.4342 


2.4545 


2.4750 


22 


68 


2.4750 


2.4959 


2.5171 


2.5386 


2.5604 


2.5826 


2.6050 


21 


69 


2.6050 


2.6279 


2.6510 


2.6746 


2.6985 


2.7228 


2.7474 


20 


70 


2.7474 


2.7725 


2.7980 


2.8239 


2.8502 


2.8770 


2.9042 


19 


71 


2.9042 


2.9318 


2.9600 


2.9886 


3.0178 


3.0474 


3.0776 


18 


72 


3.0776 


3.1084 


3.1397 


3.1715 


3.2040 


3.2371 


3.2708 


17 


73 


3.2708 


3.3052 


3.3402 


3.3759 


3.4123 


3.4495 


3.4874 


16 


74 


3.4874 


3.5260 


3.5655 


3.6058 


3.6470 


3.6890 


3.7320 


15 


75 


3.7320 


3.7759 


3.8208 


3.8667 


3.9136 


3.9616 


4.0107 


14 


76 


4.0107 


4.0610 


4.1125 


4.1653 


4.2193 


4.2747 


4.3314 


13 


77 


4.3314 


4.3896 


4.4494 


4.5107 


4.5736 


4.6382 


4.7046 


12 


78 


4.7046 


4.7728 


4.8430 


4.9151 


4.9894 


5.0658 


5.1445 


11 


79 


5.1445 


5.2256 


5.3092 


5.3955 


5.4845 


5.5763 


5.6712 


10 


80 


5.6712 


5.7693 


5.8708 


5.9757 


6.0844 


6.1970 


6.3137 


9 


81 


6.3137 


6.4348 


6.5605 


6.6911 


6.8269 


6.9682 


7.1153 


8 


82 


7.1153 


7.2687 


7.4287 


7.5957 


7.7703 


7.9530 


8.1443 


7 


83 


S.1443 


8.3449 


8.5555 


8.7768 


9.0098 


9.2553 


9.5143 


6 


84 


9.5143 


9.7881 


10.078 


10.385 


10.711 


11.059 


11.430 


5 


85 


11.430 


11.826 


12.250 


12.706 


13.196 


13.726 


14.300 


4 


86 


14.300 


14.924 


15.604 


16.349 


17.169 


18.075 


19.081 


3 


87 


19.081 


20.205 


21.470 


22.904 


24.541 


26.431 


28.636 


2 


88 


28.636 


31.241 


34.367 


38.188 


42.964 


49.103 


57.290 


1 


89 


57.290 


68.750 


85.939 


114.58 


171.88 


343.77 


00 







60 


50" 


40' 


30' 


20' 


10' 


0' 


Beg. 



NATURAL COTANGENT. 



34 



BROWN & SHARPE MFG. CO. 



CHAPTER. IV. 

WORM AND WORM WHEEL. 

(Fig. 8.) 




PROVIDENCE, R. I. 35 



FORMULAS. 

L = lead of worm. 
N = number of teeth in gear. 
. vi = threads or turns per inch in worm, 
d — diameter of worm. 
d' = diameter of hob. 
T = throat diameter. 
B = blank diameter (to sharp corners). 
C = distance between centers. 
o = thickness of hob-slotting cutter. 
/= width of lands at bottom. 
b — pitch circumference of worm. 
v = width of worm thread tool at end. 
w = width of worm thread at top. 
P = diametral pitch. 
P' = circular pitch, 
j- = addendum and module. 
t = thickness of tooth at pitch line. 
t n = normal thickness of tooth. 
/= clearance at bottom of tooth. 
D" = working depth of tooth. 
D" + / = whole depth of tooth. 

d = angle of tooth of worm wheel with its axis, or the 
angle of thread of worm with a line at right angles to its axis. 
If the lead is for single, double, triple, etc., thread, then 
Iv = P', 2 F, 3 P', etc. 



36 BROWN & SHARPE MFG. CO. 



a = 6o° to 90 

L = l 

m 

p,_ 7Tl 



D = 



N + 2 
NP __ N 
^r P 



T - _ + 2 j 

b — 7i {d — 2^) 

tan d = _ -I Practical onl y when width of wheel on wheel pitch circle 
~b \ is not more than */ z pitch diameter of worm. 

/ n = /COS 6 

1 d 

r — - — 2 s 

2 



r' 



= r' + D" +/ 



C — — s 

2 

B = T 4- 2 (r 1 — r 1 COS -1 A measurement of sketch is generally 
\ 2/ sufficient. 

„ = -335 P' + ," 

2 

- d' = d + 2/ 

z/ = .3iP' 

w = -335 p ' 



Note. — The notations and formulas referring- to tooth parts, given on page 5 for 
spur gears, apply to worm wheels, and are here used. 

Note. — Hob and worm should be marked, as per example : 
4 turns per i - ' single .25 P ; .25 L. 
'2 turns per 1" double .25 I"; .50 L. 



PROVIDENCE, R. I. 



37 



UNDERCUT IN WORM WHEELS. 

In worm wheels of less than 30 teeth the thread of the worm 
(being 29 ) interferes with the flank of the gear tooth. Such 
a wheel finished with a hob will have its teeth undercut. To 
avoid this interference two methods may be employed. 

First Met/iod. — Make throat diameter of wheel 

N 



T = cos 2 14^ 



+ 4'J 



or 



•937 N 



P 



- + 4s 



This formula increases the throat diameter, and conse- 
quently the center distance. The amount of the increase can 
be found by comparing this value of T with the one as obtained 
by formula on page 36. To keep the original center distance, 
the outside diameter of the worm must be reduced by the 
same amount the throat diameter is increased. 

Second Method. — Without changing any of the dimensions 
we found by the formulas given on page 36, we can avoid the 
interference to be found in worm wheels of less than 30 teeth 
by simply increasing the angle of worm thread. We find the 
value of this angle by the following formula : 

Let there be 

2 y = angle oi worm ihreaa. 

N = number of teeth in worm wheel. 



cos y = J ! _ jL 

y N 



From this formula we obtain the following values : 

N 



29 

3°X 


28 
3i 


27 
3i# 


26 

32X 


25 

32% 


24 23 

33^34^ 


22 
35 


21 
36 



20 



37 



2 y 



38 I 39 ! 40 ;4I^ 



15 ! 14 


13 


42^44^ 


46 % 



] 2 



48 



As this latter formula involves the making of new hobs in 
many cases, on account of change of angle, we prefer to reduce 
the diameter of worm as indicated by first method, if the dis- 
tance of centers must be absolute. 



5ROWN & SHARPS MFG. CO. 



P 

< 

— 



i4 
C 

H 

CO 



to 



£ 



C 

c 



C 

c 

GO 

w 

c 

< 

c 

P-l 

i-l 



1 = 
































CD 
































O 
































CD 
































;o 
































o 
































CD 


- -i 






























CD 
































































;o 
































































o 


ii n 






























to 
































































ico 
































































CO 


r: ■* 






























CD 
































; cd 
































o 
































o 

CD 


^H 






























































































-CD 


-n|-r 


























- 





O 
CO 


« 


























"CD 


10 


"CD 
O 


.. 






















.0 


v 




r^- 


























CO 






•*: 


f=l 






















; o 


: i 


"tO 


c 1 

tO 


"o 


























































































CO 






O 




en 


iH 






















_ 


10 


to 


"to 




o 


».H 


















S 


V- 




CO 




"0 


"cm 


CD 


« 


























































■*r 




















































tO 


CO 


C3 CN 




■* 


r~- 


CO 






































,H 
















































c; 


iO O 






■* 


•«r 






































CD 
CD 

to 


















CD 












O 


CO 


51 












: ,^_ 


CD "CD 


CO 


iO 


CD 




: 1 

-a- 


''mS- 


to 
: 1 

CO 


■s 

CO 
































V 


"3 





























IO 




5 


51 








CD 


V. 


CM 
O 


tO CM 




CO ^ 




r 

CO 


co 

CO 


CM 
"CO 


-T 

CO 


-to 

CO 

CM 














- 


* 






















CN 







>* 


O CM 


CO 









CO 






SI 




"0 


tO 


"-1. 




-T 

CD 


"* : — 


"■^ 


c 4 1 c 4- 


CO 


CO 


CO 


CO 


CO 




































CD 






CO 












O 




co 


CD 








IO 






CO 




























s 


51 




: 1 


: ' 


- < 




Z 1 






: 1 










: 1 








^ 


CD 


CO 


r- 


CD 


IO 


LO 


"-■ 


-^ 


■* 




CO 


CO 


CO 




CM 










- 


- 
































CO 




CO 


- 


l-~ 


CO 


CO 






CD 


to 






















CM 


to 


CO 
















CD 


51 


o 


CD 


I-- 


a 


~o 


" 1 


C ' 


: 4 


"«1 


~o CO 


CO 


CO 


CO 


"cm 


cm 












































CM 





















r^ 














CO 


CO 








CM 




■* 








to 






CO 


51 


o 


"i 


°li 


CI 


a 


LTD 




r 4 


^* 


"co 


"co 


: co 


CO 


CM 


"cm 


CM 


- CO 




"co 

CD 


„ 


o 


» 






« 


- 


%r 




\_ 


^ 


CM 


CM 




"cd 


c~ 


ee 


CO 


tO 




0: 


tO 


CM co 


•* 


CI 


■— 


CM 


to 


"* 


CO 


CM 


CO 






"cO 


CD 





"to 




^ -^ 


"co 




"co 


CO 


Cs 


CM 


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PROVIDENCE, R. I. 



39 



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40 



BROWN & SHARPE MFG. CO. 



CHAPTER V 



SPIRAL OR SCREW GEARING. 

(Figs. 9, 10, ii. 




Fig. 9. 

RIGHT HAND SPIRAL GEARS. 

In spiral gearing the wheels have cylindrical pitch surfaces, 
but the teeth are not parallel to the axis. The line in which 
the pitch surface intersects the face of a tooth is part of a 
screw line, or helix, drawn at the pitch surface. A screw 
wheel may have one or any number of teeth. A one-toothed 
wheel corresponds to a one-threaded screw, a many-toothed 
wheel to a many- threaded screw. The axes may be placed at 
any angle. 

Consider spiral gears with : 

I. Axes parallel. 
II. Axes at right angles. 
III. Axes any angle. 



PROVIDENCE, R. I. 



4* 




Let th 

N a 
N ft 
C 
P 
P' 
P" 

Y 
L, 

L 2 

T 

D 
B 



Fig. 10. 

LEFT HAND SPIRAL GEAR. 

ere be : 

~ > number of teeth in gears 

= center distance. 

— diametral pitch 

= circular pitch. 

= normal diametral pitch. 

== normal circular pitch. 

= angle of axes. 

= exact lead of spiral on pitch surface. 

= approximate lead of spiral on pitch surface. 

= number of teeth marked on cutter to be used when 

teeth are to be cut on milling machine. 
= pitch diameter. 
= blank diameter. 



a n — 



t 

s 



angle of teeth with axis 

= thickness of tooth. 

= addendum and module. 

= whole depth of tooth. 



D"+/ 

Note. — Letters a and b occurring- at bottom of notations refer to gears a and b. 

I. — Axes Parallel. 

Gears of this class are called twisted gears. The angle of 
teeth. with axes in both gears must be equal and the spirals 
run in opposite directions. The angles are generally chosen 
small (seldom over 20 ) to avoid excessive end thrust. End 
thrust may, however, be entirely avoided by combining two 
pairs of wheels with right and left-hand obliquity. Gears of 
this class are known as Herringbone gears. They are com- 
paratively noiseless running at high speed. 



42 BROWN & SHARPE MFG. CO. 

II. — Axes at Right Angles. 

Here we must always have : 

i. The teeth of same hand spiral ; 

2. The normal pitches equal in both gears ; and 

3. The sum of the angles of teeth with axes = 90°. 

Choosing Angle of Teeth with Axes. 

1. If in a pair of gears the ratio of the number of teeth is 
equal to the direct ratio of the diameters, i. e., if the number of 
teeth in the two gears are to each other as their pitch diame- 
ters, then the angles of the spirals will be 45° and 45° ; for, this 
condition being fulfilled, the circular pitches of the two gears 
must be alike, which is only possible with angles of 45°. In 
such a combination either gear may be the driver. 

2. If the ratio of the diameters determined upon is larger 
or smaller than the ratio of the number of teeth, then the 
angles are : 

tan a a = ^^ tan a b = ° b Nq 
D 6 N a D N\ 

In such gears the velocity ratio is measured by the number 
of teeth, and not by the diameters. 

3. Given N a , N 6 and C : 

If P a ' is made = P 6 ', then we have case " 1 " and 

P' — n ^ 
~^(N a +N 6 ) 

But if P a ' is assumed, then : 

p r C n—yi N a P a ' 

b = i/^NJ 

and 

tan a n = — ± tan a b = — - 

The gear whose P' or a is larger will ordinarily be the 
driver, on account of the greater obliquity of the teeth. 

4. Given N a , N 6 and C or D. 
See case " 7 " under III., considering y = 90 . 

III. — Axis at any Angle (y). 

5. Given case " 1," under II., then angles of spirals — j4 y, 
for the same reason. 

6. Analogous cases to "2" and "3," under II., may be 
worked out, when angles of axes = y, but they have been 



PROVIDENCE, R. I. 



43 



omitted, partly because the formulas are too cumbersome, and 
partly because they are to some extent covered by cases "5 " 

and "7." 

7. Given N a , N b and C, or one of the pitch diameters. We 
find the angles by a graphic method, which for all practical 
purposes is accurate enough ; ro and v o are the axes of gears 
forming angle y (see diagram, Fig. 11.) On these axes we 
lay off lines o r and v representing the ratio of the number 
of teeth (velocity ratio), so that N a : N 6 : : r s : s v, and 




Fig. 11. 



construct parallelogram o r s v. Then, according to Mc- 
Cord,* the angles formed by the tangent s o in the pitch con- 
tact with the axes of the gears insures the least amount of 
sliding. In bisecting angle y by tangent u o and using angles 
produced in this manner we equally distribute the e?id thrust on 
both shafts. Both methods have their advantages ; to profit 
by both we select angles a a and a h , produced by tangent x, 
bisecting angle u s. 

Thus we have when angles are found and C given, 

2 C 7t cos a a cos al 



p/>i 



and when D Cf given 



p/»i 



D, 



N a cos a b + Nfccosa^ 
D„ 7t cos a„ 



P' n N 6 
it cos a h 



and 



* McCord, Kinematics, page 378. 



44 



EROWX & 5HARPE MFG. CO. 



General Formulas. 



y = a a + a , 

n ':: "D 'n 

D = or = 

B = D + 2J 



P = or = 

X 



-tcos a 
or = D - 

P 



P 



cos a 



P = 
P = 



P cos a 

7T 



t = 



F 

pn. 

T 
?' 



or 



i Piich of cutter.) 
r 



D "-/= 2 j + 

L : = 

L = 



cos a 

X P' 
tan a 

ic WG. 



i ->::' J\~<?fe 7.) 



:r 



x- 



:: 



, L : = X P'i 



S G 



Ptan«r • L; = X" : P' ; 

- Note 2 and examph 



: s 45°= 7071 1 

( cos 45= : 5 " : 
* tan 45 : = 1.000 

Note i. — Cutters of regular involute system. 



L'sr No. 1 cutter for T from 135 up. 

2 55 *-o 134 

■• : • ■ $5 to 54 

' 4 26 to 34 



No. 5 cutter for T from 

" 6 " 

. . 



21 te 

17 t: _ 
14 to 16 

12 tO 13 



Note 2. — Gears used on spiral head and bed for Brown 3l Sharpe milling 

machine : 

W = number of teeth in gear on worm. 

G = •• is: •• d. 

G: = •• •• 2d • stud. 

5 = •• •• screw. 

Should a spiral head of different construction be used, the formula might not 
apply. 



PROVIDENCE, R. I. 



45 



The following data are usually required in cutting spiral 
gears in a Universal Milling Machine, and it will be found 
convenient to arrange them in tabular form as follows : 



' 1 


GEAR. 


PINION. 


No. of Teeth --------- 

Pitch Diameter --------- 

Outside Diameter - -- - - - 

Circular Pitch --------- 

Angle of Teeth with Axis ----- 

Normal Circular Pitch ------ 

Pitch of Cutter - - - - 

Addendum s --------- 

Thickness of Tooth t ------ 

Whole Depth D" + f - - - - - - - 

No. of Cutter --------- 

Exact Lead of Spiral ------- 

Approximate Lead of Spiral - - - - 






Gears on Milling Machine to Cut Spiral 
Gear on W T orm --------- 

ist Gear on Stud -------- 

2nd Gear on Stud -------- 

Gear on Screw --------- 







If the exact lead L x can be obtained by the gears at hand, 
Lj will equal L 2 and we shall have from the formula 
10 W Go 

,L<o — 



S G L 

W Go 

-g-g^ (for B. & S. Milling Machine.) 



10 

Example I. 

Required the gears for cutting a spiral of 2%" lead. 

-— = - factoring, in the most simple way, we have 

i i x i i x 28 32 x 28 W G 2 
4 - ~2X2~"56x2~"56 x~64 S G! 



\ 



46 BROWN & SHARPK MFG. CO. 

Thus the gearing will be 32 T.. on worm, 64 T. ist, on stud, 
28 T. 2nd on stud 3 and 56 T. on screw. 

Trying these gears on the Milling Machine we find that 
they cannot be used, and as we have no other regular gears 
in the ratio of 2 to 1 that can be used we must try, by factor- 
ing, to get such ratios for the two pairs of gears as to be able 
to use the gears at hand, bearing in mind that the combined 
ratio must be J. 

1 18 3x6 24 x 6 24 x 48 
4~~~ 72 ~~9x8 __ 9x 64 ~~ 72 X64 

These gears are at hand and the combination can be used 
on the machine, giving the exact lead of 2J". 

Example II. 

Required the gears for cutting a spiral of 8.639" lead. 

8.639 = 8 T 6 3 (H) ; reducing, by continued fractions, to a 
smaller fraction of approximately the same value, as described 
on pages 74 and 75 



639 ) 1000 ( 1 
639 

361 )639( 1 
361 

278 ) 361 ( 1 
278 

~&3) 278 ( 3 
J>49_ 

29 ) 83 ( 2 
58 

25 ) 29 ( 1 
25 

4)25 (6 
24 

i)4(4 
4 



PROVIDENCE, R. I. 47 



112. _7_ 1.6 .2 3. J_5.4 6 3!) 
1 2 3 11 2 5 3 6 2 4 110 TJTT 

Selecting ^-f as an approximation near enough for our 
purpose, and in fact as near as we are likely to find gears for, 
we have for our lead 8i|. Applying the formula as in Ex- 
ample I. 

Sjj __ W G, 
io S Gi 

8||- 216 108 f 

— — = — — = — - factoring we have 
io 250 125 & 

9 X 12 9 X 48 72 X 48 . 

^r — z = T ^ „ : = 7Z~: 7: the gears required, 

. 25 x 5 100 x 5 100 x 40 & H 

these being regular gears furnished with the Milling Machine. 

Proof : 

72 x 48 x 10 

- = 8.640 = L, 

100 X 40 , T " 

8-639 = Li 
.001" error in lead. 

In shops where much work is done in milling spirals it is 
desirable to have a full set of gears for the milling machine, 
from the smallest to the largest numbers of teeth that can be 
used. This makes it possible, in most cases, to get closer 
approximations than could be otherwise obtained, and often 
saves a great deal of figuring. 

When the use of continued fractions does not bring a 
close enough approximation, one method to secure a closer 
result is to add to or substract from the numerator and de- 
nominator of the fraction to be reduced, any numbers nearly 
in proportion to the given fraction, seeing that the numbers 
added or substracted are such as to make the fraction reduc- 
ible to lower terms. By a little ingenuity and patience ex- 
tremely close approximations can generally be reached in 
this way. 

Take, as an illustration, the fraction in Example II. 

8tVo 9 o 8639 



10 1 0000 

Adding 9 to the numerator and 10 to the denominator, these 



48 BROWN & SHARPE MFG. CO. 

being in about the same ratio to each other as the numerator 
and denominator of the fraction, we have 

8639+9 = 8648 __ 4324 = _ 47 x 92 
10000+10 = 10010 ~ " 5005 55 x 91 

All of the gears in this case are special. 

Applying the same proof as in Example II. we find that 
this train of gears will give a lead of 8.6393+, making an 
error of .0003" in the lead. 

Xo doubt a much closer approximation even than this 
could be obtained by further trial. 

Another method is to multiply both terms of the fraction 
by some number which will make one term of the fraction 
easily reducible, and adding one to or subtracting it from the 
other term to make it possible to reduce that also. 

There is an element of uncertainty in both these methods, 
as we never feel sure that we have obtained the best combina- 
tion ; practical work, however, rarely requires accuracy beyond 
a point that can readily be reached. 

The accompanying list of prime numbers and factors will 
be found useful in reducing: and factoring fractions. 



PROVIDENCE, R. I. 



49 



PRIME NUMBERS AND FACTORS. 
1 TO lOOO, 



1 




26 


2x13 


51 


3x17 


76 


■ 1 

2 2 xl9 


2 




27 


3' 


52 


2 2 xl3 


77 


7x11 


3 




28 


2 2 x7 


53 




78 


2x3x13 


4 


2 2 


29 




54 


2x3" 


79 









30 


2x3x5 


55 


5x11 


80 


2 4 x5 


6 


2x3 


31 




56 


2' x 7 


81 


3 4 


7 




32 


2" 5 


57 


3x 19 


82 


2x41 


8 


2 3 


33 


3x11 


58 


2x29 


oo 




9 


3- 


34 


2x17 


59 




84 


2 2 x 3 x 7 


10 


2x5 


35 


5x7 


60 


2 2 x 3 x 5 


85 


5x 17 


11 




36 


2 2 x3 2 


61 




86 


2x43 


12 


2 2 x3 


37 




62 


2x31 


87 


3 x 29 


13 




38 


2x19 


63 


3 2 x7 


88 


2 3 xll 


14 


2x7 


39 


3x13 


64 


2° 


89 




15 


3x5 


40 


2 3 x5 


65 


5x13 


90 


2x3 2 x5 


16 


2 4 


41 




66 


2x3x11 


91 


7x13 


17 




42 


2x3x7 


67 




92 


2 2 x23 


18 


2x3 2 


43 




68 


2 2 xl7 


93 


3x31 


19 




44 


2 2 xll 


69 


3x23 


94 


2x47 


20 


2 2 x5 


45 


3 2 x5 


70 


2x5x7 


95 


5x19 


21 


3x7 


46 


2x23 


71 




96 


2 5 x3 


22 


2x11 


47 




72 


2 3 x3 2 


97 




23 




48 


2 4 x3 


73 




98 


2x7 2 


24 


2 3 x3 


49 


f-9 

r 


74 


2x37 


99 


3 2 xll 


25 


5 2 


50 


2x5 2 


75 


3x5 2 


100 


2 2 x5 2 



5o 



BRCTSYX & SHARPE MFG. CO. 



101 




131 




161 


7 x 23 


191 


= 


102 


2x3x17 


132 


2-x3xll 


162 


2x3 4 


192 


2'x3 


103 




133 


7x19 


163 




193 




104 


2°xl3 


134 


2 x 67 


164 


2 2 X41 


194 


2x97 


105 


3x5x7 


135 


3 3 x 5 


165 


3 x 5 x 11 


195 


3x5x13 


106 


2x53 


136 


2 3 xl7 


166 


2 x 83 


196 


2-x7- 


107 




137 




107 




197 




108 


2-x3 3 


138 


2 X 3 x 23 


168 


•?■ x 3 x 7 


198 


2x3^x11 


109 




139 




169 


13- 


199 




110 


2X5X11 


140 


2- x 5 x 7 


i7<:> 


2x5x17 


200 


2 3 x5 2 


111 


3x37 


141 


3x47 


171 


3-xl9 


201 


3x67 


112 


2 4 x7 


142 


2x71 


172 


2 2 x43 


202 


2x101 


1J3 




143 


11x13 


173 




203 


7x29 


114 


2x3x19 


144 


2 4 x3^ 


174 


2x3x29 


204 


2-x3xl7 


115 


5x23 


145 


5 x 29 


175 


5- x 7 


205 


5x41 


116 


2-X29 


146 


2 x 73 


176 


2 4 xll 


206 


2 x 103 


117 


3-X13 


147 


3 x 7- 


177 


3 x 59 


207 


3^x23 


118 


2x 


148 


2 2 x37 


178 


2x89 


2 U.s 


2 4 xl3 


119 


7x17 


149 




179 




. 


11 xl9 


120 


2 s x 3 x 5 


150 


2 x 3 x 5- 


180 


2 2 x 3- x 5 


210 


2x3x5x7 


121 


11- 


151 




181 




211 




122 


2x61 


152 


2 3 xl9 


182 


2x7x 13 


212 


2-X53 


123 


3x41 


153 


3 J xl7 


183 


3x61 


213 


3x71 


124 


2-X31 


154 


2x7x11 


184 


2-X.23 


214 


2x107 


125 


5 3 


155 


5x31 


185 


5x37 


215 


5x43 


126 


2x3^x7 


156 


2 2 X3X13 


186 


2x3x31 


216 


2 3 x3 8 


127 




157 




187 


11 X17 


217 


7x31 


128 


2" 


158 


2x79 


188 


2-x47 


218 


2 x 109 


129 


3 x -43 


159 


3 x 53 


189 


3 $ x 7 


219 


3 x 73 


130 

i 


2x5x13 


160 


2 s x 5 


19(> 


2x5x19 


220 


2- x 5 x 1 1 



PROVIDENCE, R. I. 



51 



1 

221 


13x17 


251 




281 




311 




222 


2x3x37 


252 


2 2 x3 2 x7 


282 


2x3x47 


312 


2 3 x3xl3 


223 




253 


11x23 


283 




313 




224 


2 5 x7 


254 


2X127 


284 


2 2 x71 


314 


2x157 


225 


3 2 x5 2 


255 


3x5x17 


285 


3x5x19 


315 


3 2 x5x7 


226 


2x113 


256 


2 8 


286 


2x11x13 


316 


2 2 x79 


227 




257 




287 


7x41 


317 




228 


2 2 x3xl9 


258 


2 X 3 X 43 


288 


2 5 x3 2 


318 


2x3x53 


229 




259 


7x37 


289 


17 2 


319 


11x29 


230 


2x5x23 


260 


2 2 x5xl3 


290 


2x5x29 


320 


2 6 x5 


231 


3X7X11 


261 


3 2 x29 


291 


3x97 


321 


3x107 


232 


2 3 x29 


262 


2x131 


292 


2 2 x73 


322 


2x7x23 


233 




263 




293 




323 


17x19 


234 


2x3 2 xl3 


264 


2 3 x3xll 


294 


2 X 3 x 7 2 


324 


2 2 x3 4 


235 


5 x47 


265 


5 x53 


295 


5x59 


325 


5 2 X13 


236 


2 2 x59 


266 


2x7x19 


296 


2 3 x37 


326 


2x163 


237 


3x79 


267 


3x89 


297 


3 3 xll 


327 


3x109 


238 


2x7x17 


268 


2 2 X67 


298 


2x149 


328 


2 3 x41 


239 




269 




299 


13x23 


329 


7x47 


240 


2 4 x3x5 


270 


2x3 3 x5 


300 


2 2 x3x5 2 


330 


2X3X5X11 


241 




271 




301 


7x43 


331 




242 


2xll 2 


272 


2 4 xl7 


302 


2x151 


332 


2 2 x83 


243 


r> 5 


273 


3x7x13 


303 


3x101 


333 


3 2 x37 


244 


2 2 X61 


274 


2x137 


304 


2 4 X19 


334 


2x167 


245 


5X7 2 


275 


5 2 Xll 


305 


5x61 


335 


5x67 


246 


2x3x41 


276 


2 2 x3x23 


306 


2x3 2 xl7 


336 


2 4 x3x7 


247 


13x19 


277 




307 




337 




248 


2 3 x31 


278 


2x139 


308 


2 2 x7xll 


338 


2X13 2 


249 


3x83 


279 


3 2 x31 


309 


3x103 


339 


3x113 


250 

. 


2x5 3 


280 


2 3 x 5 X 7 


310 


2x5x31 


340 


2 2 X5X17 



BROWN & SHARPE MFG. CO. 



341 


11x31 


371 


7x53 


401 


431 


, . =:_ 


342 


2x3 2 Xl9 


372 


2 2 x3x31 


402 2x3x67 


432 


2 4 x3 :i 


343 


7 ,J 


373 




403 13x31 


433 




344 


2^X43 


374 


2x11x17 


404 


2 2 xl01 


434 


2x7x31 


345 


£ X5 x23 


375 


3 X 5 s 


|405 


3 4 x5 


435 


3 X 5 X 29 


346 


2x173 


376 


2 3 x47 


406 2x7x29 


436 


2 2 xl09 


347 




377 


13x29 


407 


11x37 


437 


19x23 


348 


2 2 X 3 x 29 


378 


2 x 3 3 x 7 


408 


2 3 x3xl7 


438 


2x3x73 


349 




879 




409 


439 




350 


2x5-x7 


380 


2 2 x5xl9 


410 2x5x41 


440 


2>x5x 11 


351 


3 3 X13 


381 


3x127 


411 3x137 


441 


3 2 x 7 2 


352 


2 5 Xll 


382 


2X191 


412 2 2 xl03 


412 


2x13x17 


353 




883 


i 


413 7x59 


443 




354 


2 X 3 X 59 


384 


2 X o 


414 2x3 2 x23 


444 


2 2 x3x37 


355 


ox 71 


385 


5x7x11 


415 5x83 


445 


5 X 89 


356 


2 2 X 89 


386 


2x193 


416 2'xl3 


446 


2 x 223 


357 


3x7x17 


387 


3 2 x43 


417 


3x139 


447 


3x149 


358 


2x179 


388 


2 2 x97 


418 


2x11x19 


448 


2 6 x7 


359 




389 




419 


__ 


449 




360 


2-x3 2 x5 


390 


2X3X5X13 


420 


2 2 X3X5X7 


450 


2x3 2 x5 2 


361 


19 2 


391 


17x23 


421 




451 


11X41 


362 


2x181 


392 


2 3 X7 2 


422 


2x211 


452 


2 2 xH3 


363 


3xll 2 


393 


3x131 


423 


3 2 x47 


453 


3x151 


364 


2 2 x7xl3 


394 


2x197 


424 


2 3 X53 


454 


2x227 


365 


5x73 


395 


5 x 79 


425 


5 2 x 1 7 


455 


5x7x13 


366 


2x3x61 


396 


2 2 x3 2 xll 


426 


2x3x71 


456 


2 ; x3xl9 


367 




397 




427 


7x61 


457 




368 


2 4 x 23 


398 


2x199 


428 


2 2 X107 


458 


2x229 


369 


3 2 x41 


399 


3x7x19 


429 


3xllXl3 


459 


3 3 xl7 


370 




! 2x5x37 


400 


2 4 X5 2 


430 


2x5x43 


460 


2 2 x5x23 



PROVIDENCE, R. I. 



461 




491 




521 




551 


19x29 


462 


2X3X7X11 


492 


2 2 x3x41 


522 


2x3 2 x29 


552 


2 3 x3x23 


463 




493 


17X29 


523 




553 


7x79 


464 


2 4 X 29 


494 


2x13x19 


524 


2 2 xl31 


554 


2x277 


465 


3x5x31 


495 


3 2 x5xll 


525 


3 x 5 2 x 7 


555 


3x5x37 


466 


2 X 233 


496 


2 4 x31 


526 


2x263 


556 


2 2 xl39 


467 




497 


7x71 


527 


17x31 


557 




468 


2 2 x3 2 xl3 


498 


2x3x83 


528 


2 4 x3xll 


558 


2x3 2 x31 


469 


7x67 


499 




529 


23 2 


559 


13x43 


470 


2x5x47 


500 


2 2 x 5 3 


530 


2x5x53 


560 


2 4 x5x7 


471 


3x157 


501 


3x167 


531 


3 2 X59 


561 


3x11x17 


472 


2 3 x59 


502 


2x251 


532 


2 2 x7xl9 


562 


2x281 


473 


11x43 


503 




533 


13x41 


563 




474 


2x3x79 


504 


2 3 x3 2 x7 


534 


2 x 3 x 89 


564 


2 2 x3x47 


475 


5 2 X19 


505 


5x101 


535 


5x107 


565 


5x113 


476 


2 2 x 7 X 1 7 


506 


2x11x23 


536 


2 3 x67 


566 


2x283 


477 


3 2 x53 


507 


3xl3 2 


537 


3x179 


567 


3 4 x7 


478 


2 x 239 


508 


2 2 xl27 


538 


2 x 269 


568 


2 3 X71 


479 




509 




539 


7 2 xll 


569 




480 


2 5 x 3 x 5 


510 


2X3X5X17 


540 


2 2 x3 3 x5 


570 


2x3X5X19 


481 


13.X37 


511 


7x73 


541 




571 




482 


2x241 


512 


2 9 


542 


2x271 


572 


2 2 xllxl3 


483 


3 x 7 x 23 


513 


3'X19 


543 


3x181 


573 


3x191 


484 


2 2 xll 2 


514 


2x257 


544 


2 5 xl7 


574 


2x7x41 


485 


5x97 


515 


5x103 


545 


5x109 


575 


5 2 x23 


486 


2 x 3 5 


516 


2 2 x3x43 


546 


2X3X7X13 


576 


2 6 x3 2 


487 




517 


11x47 


547 




577 




488 


2 s x61 


518 


2x7x37 


548 


2 2 xl37 


578 


2xl7 2 


489 


3x163 


519 


3x173 


549 


3 2 x61 


579 


3x193 


490 


2 x 5 x 7 2 

— _ L 


520 


2 3 x5xl3 


550 


2x5 2 xll 


580 


2 2 x5x29 



54 



BROWN & SHARPE MFG. CO. 



581 


7x83 


611 13x47 


1 

611 671 


11x61 


582 


2x3x97 


612 2 2 x3 2 xl7 


642 


2x3x107 


672 


2 5 x 3 x 7 


583 


llx-33 


613 




643 




673 




584 


2 a x73 


614 


2x307 


644 


2 2 x7x23 


674 


2x337 


585 


3-x5xl3 


615 


3x5x41 


645 3x5x43 


075 


3 3 X5 2 


586 


2 x 293 


616 


2 3 x7xll 


646 


2x17x19 


676 


2 2 xl3 2 


587 




617 




617 




677 




588 


2 2 x 3 x 7 2 


618 


2x3x103 


648 


2'x3 4 


678 


2x3x113 


589 


19x31 


619 




649 


11X59 


679 


7x97 


590 


2 x5x59 


020 


2 2 xox31 


650 


2x5 2 xl3 


680 


2x5 x 17 


591 


3x197 


621 


3 ;; x23 


651 


3x7x31 


681 


3x227 


592 


2 4 x37 


622 


2x311 


652 


2 2 xl63 


682 


2x11x31 


593 




623 


7x89 


653 




683 




594 


2 x 3 ■'■ x 1 1 


624 


2 4 x3xl3 


654 2x3x109 


684 


2 2 x3 2 xl9 


595 


5x 7 x 17 


625 


5 4 


655 5x131 


685 


5x137 


596 


2 2 xl49 


626 


2x313 


656 2 4 x41 


686 


2 x 7 • 


597 


3 x 199 


627 3x11x19 


657 3 2 x73 


687 


3x229 


598 


2x13x23 


628 2 2 xl57 


658 2 x 7 x47 


688 


2 4 x43 


599 




629 17x37 


659 




689 




600 


2 s x 3 x 5 2 


630 


2X 3"X 5X7 


660 


2 2 X3X5xll 


690 


2X3x5X23 


601 




631 




661 




691 


• 


602 


2x7x43 


632 


2'x79 


662 


2x331 


692 


2 2 xl73 


603 


3-x67 


633 


3x211 


663 


3x13x17 


693 


3 2 x7xll 


604 


2-xlol 


634 


2x317 


664 


2'x83 


694 


2 x 347 


605 


5xll 2 


635 5x127 


665 5x7x19 


695 


5x139 


606 


2x3x101 


636 2 2 x3x53 


666 2x3 2 x37 


696 


2 ; x3x29 


607 




637 


72 ^ 1 Q 

1 X 1 3 


667 23x29 


697 


17x41 


608 


2 5 xl9 


638 2x11x29 


668 2 2 xl67 698 


2 x 349 


609 


3x7x21' 


639 3-X71 


669 3x223 699 


3 x 233 


610 


2x5x61 


640 2 7 X 5 


670 2x5x67 700 


2 2 x 5 2 X 7 



PROVIDENCE, R. I. 



55 



701 




731 


17x43 


761 




791 


i ■ ■ 

7x113 


702 


2x3 3 xl3 


732 


2 2 x3x61 


762 


2x3x127 


792 


2 3 x3 2 xll 


703 


19x37 


733 




763 


7x109 


793 


13x61 


704 


2 c xll 


734 


2x367 


764 


2 2 xl91 


794 


2x397 


705 


3x5x47 


735 


3 x 5 x 7 2 


765 


3 2 x5xl7 


795 


3x5x53 


706 


2x353 


736 


2 5 x23 


766 


2x383 


796 


2 2 xl99 


707 


7x101 


737 


11x67 


767 


13x59 


797 




708 


2 2 x3x59 


738 


2x3 2 x41 


768 


2 8 x3 


798 


2X3X7X19 


709 




739 




769 




799 


17x47 


710 


2x5x71 


740 


2 2 x5x37 


770 


2X5X7X11 


800 


2 5 x5 2 


711 


3 2 X79 


741 


3x13x19 


771 


3x257 


801 


3 2 x89 


712 


2 3 x89 


742 


2x7x53 


772 


2 2 xl93 


802 


2x401 


713 


23x31 


743 




773 




803 


11X73 


714 


2X3X7X17 


744 


2 3 x3x31 


774 


2x3 2 x43 


804 


2 2 x3x67 


715 


5x11x13 


745 


5x149 


775 


5 2 x31 


805 


5 X 7 X 23 


716 


2 2 xl79 


746 


2x373 


776 


2 3 x97 


806 


2x13x31 


717 


3x239 


747 


3 2 x83 


777 


3x7x37 


807 


3x269 


718 


2 x 359 


748 


2 2 xllxl7 


778 


2x389 


808 


2 3 X101 


719 




749 


7x107 


779 


19x41 


809 




720 


2 4 x3 2 x5 


750 


2 x 3 x 5 3 


780 


2 2 X3X5X13 


810 


2x3 4 x5 


721 


7x103 


751 




781 


11X71 


811 




722 


2xl9 2 


752 


2 4 x47 


782 


2x17x23 


812 


2 2 x7x29 


723 


3x241 


753 


3x251 


783 


3 3 x 29 


813 


3x271 


724 


2 2 xl81 


754 


2x13x29 


784 


2 4 x7 2 


814 


2x11x37 


725 


5 2 x29 


755 


5x151 


785 


5 x 157 


815 


5x163 


726 


2x3xll 2 


756 


2 2 x3 3 x7 


786 


2x3x131 


816 


2 4 x3xl7 


727 




757 




787 




817 


19x43 


728 


2 3 x7xl3 


758 


2x379 


788 


2 2 xl97 


818 


2x409 


729 


3 6 


759 


3x11x23 


789 3x263 


819 


3 2 X7X13 


730 
I 


2x5x73 


760 


2 3 x5xl9 


790 2x5x79 


820 


2 2 x5x41 



BROWN cS: SHARPE MFG. CO. 



821 


I 


851 


23x37 


881 


911 


1 


822 


2x3x137 


852 


2 2 x3x71 


! 882 


2x3 2 x7 2 


912 


2 4 x3xl9 


823 




853 




883 




913 


11x83 


824 


2 3 xl03 


854 


2x7x61 


884 


2 2 xl3xl7 


914 


2x457 


825 


3x5 2 xll 


855 


3 2 x5xl9 


885 


3x5x59 


915 


3x5x61 


826 


2 x 7x59 


856 


2 3 xl07 


886 


2x443 


916 


2 2 x229 


827 




857 




887 




917 


7x131 


828 


2 2 x3 2 x23 


858 


2x3x11x13 


888 


2 3 x3x37 


918 


2x3 3 xl7 


829 




859 




889 


7x127 


919 




830 


2x5x83 


860 


2 2 x5x43 


890 


2x5x89 


920 


2 3 x5x23 


831 


3x277 


861 


3x7x41 


891 


3 4 xll 


921 


3x307 


832 


2 G xl3 


862 


2x431 


892 


2 2 x223 


922 


2x461 


833 


7 2 xl7 


863 




893 


19x47 


923 


13x71 


834 


2x3x 139 


864 


2 5 x 3 3 


894 


2x3x149 


924 


2 2 x3X7XH 


835 


5xl'67 


865 


5x173 


895 


5x179 


925 


5 2 x37 


836 


2 2 xllxl9 


866 


2x433 


896 


2 7 x7 


926 


2x463 


837 


3 s x31 


867 


3xl7 2 


897 


3x13x23 


927 


3 2 xl03 


838 


2x419 


868 


2 2 x7x31 


898 


2x449 


928 


2 5 x29 


839 




869 : 


11x79 


899 


29x31 


929 




840 


2 3 X3X5X7 


870 


2X3X5X29 900 2 2 x3 2 x5 2 


930 2X3X5X31 


841 


29 2 


871 


13x67 901 17x53 


931 7 2 xl9 


842 


2x421 


872 


2 3 xl09 


902 


2x11x41 


932 2 2 x233 


843 


3x281 


873 


3 2 x97 


903 


3x7x43 


933 3x311 


844 


2 2 x211 


874 


2x19x23 


904 


2 3 xll3 


934 2x467 


845 


5X13 2 


875 


5 3 x 7 


905 


5x181 


935 5x11x17 


846 


2x3 2 x47 


876 


2 2 x 3 x 73 


906 


2x3x151 


936 


2 3 x3 2 Xl3 


847 


7xll 2 


s;; 




907 




937 




848 


2 4 x53 


878 


. 2x439 


908 


2 2 x227 


938 


2x7x67 


849 


3x283 


879 


3x293 


909 


3 2 xl01 


939 


3x313 


850 


2 X 5 2 X 1 7 880 


2 4 X0Xll 910 


2X5X7X13 


940 


2 2 x5x47 

i 



PROVIDENCE, R. I. 



o7 



941 




956 


l 
2 2 x239 


971 




986 


2x17x29 


942 


2x3x157 


957 


3x11x29 


972 


2 2 X 3 5 


987 


3x7x47 


943 


23x41 


958 


2x479 


973 


7x139 


988 


2 2 x 13x19 


944 


2 4 X59 


| 050 


7x137 


974 


2x487 


989 


23 X 43 


945 


3 3 x 5 x 7 


I960 


2 fi x3x5 


975 


3x5 2 xl3 


990 


2x3 2 X5Xll 


946 


2x11 X43 


961 


31 2 


976 


2 4 x61 


991 




947 




962 


2x13x37 


977 




992 


2 5 x31 


948 


2 2 x3x79 


963 


3 2 xl07 


978 


2x3x163 


993 


3x331 


949 


13x73 


964 2-X241 


979 


1 1 X 89 


994 


2 x 7 x 71 


950 


2X5-X19 


965 5x193 


980 


2 2 X 5 x 7 2 


995 


5x199 


951 


3x317 


966 


2X3X7X23 


981 


3 2 xl09 


996 


2 2 x3x83 


952 


2 3 x7xl7 


967 


i 


982 


2x491 


997 




953 




968 


2'Xll 2 


983 




998 


2x499 


954 


2x3 2 x53 


969J 3x17x19 


984 


28 x 3x41 


999 


3 3 x37 


955 

c- .- - 


5x191 


970 2x5x97 


985 


5x197 


1000 

i 


2 3 x 5 3 



58 BROWN & SHARPE MFG. CO. 



CHAPTER VI. 



INTERNAL GEARING. 

PART A.— INTERNAL SPUR GEARING. 
(Figs. 12, 13, 14, 15, 16.) 

A little consideration will show that a tooth of an internal 
or annular gear is the same as the space of a spur — external 
gear. 

We prefer the epicycloidal form of tooth in this class of 
gearing to the involute form, for the reason that the difficulties 
in overcoming the interference of gear teeth in the involute 
system are considerable. Special constructions are required 
when the difference between the number of teeth in gear and 
pinion is small. 

In using the system of epicycloidal form of tooth in which 
the gear of 15 teeth has radial flanks, this difference must be 
at least 15 teeth, if the teeth have both faces and flanks. Gears 
fulfilling this condition present no difficulties. Their pitch 
diameters are found as in regular spur gears, and the inside 
diameter is equal to the pitch diameter, less twice the adden- 
dum. 

If, however, this difference is less than 15, say 6, or 2, or 1, 
then we may construct the tooth outline (based on the epicy- 
cloidal system) in two different ways. 

First Method. — To explain this method better, let us sup- 
pose the case as in Fig. 12, in which the difference between 
gear and pinion is more than 15 teeth. Here the point o of 
the describing circle B (the diameter of which in the best 
practice of the present day is equal to the pitch radius of a 15 
tooth gear, of the same pitch as the gears in question) gene- 
rates the cycloid o, o 1 , o a , o 3 , etc., when rolling on pitch circle 
L L of gear, forming the face of tooth ; and when rolling on 
the outside of L L the flank of the tooth. In like manner is the 
face and flank of the pinion tooth produced by B rolling out- 
side and inside of E E (pitch circle of pinion). A little study 



PROVIDENCE, R. I. 



59 



of Fig. 12 (in which the face and flank of a gear tooth are 
produced) will show the describing circle B divided into 12 



„-'S 




^ 




» «0 




«fc 


*> 


© 


>H 





# 




fe 




IS* 




^ 




equal parts and circles laid through these points (1, 2, 3, etc.), 
concentric with L L. We now lay off on L L the distances 
0-1, 1-2, 2-3, etc., of the circumference of B, and obtain points 



6o 



BROWN & SHARPE MFG. CO. 



i 1 , 2 1 , 3 1 , etc. [Ordinarily it is sufficient to use the chord.] It 
will now readily be seen that B in rolling on L L will success- 
ively come in contact with i 1 , 2 1 , 3 1 , etc., c meanwhile moving 
to c\ r, c\ etc. (points on radii through i 1 , 2 1 , 3 1 , etc.), and the 
generating point o advancing to o 1 , o 2 , o 3 , etc., being. the inter- 
sections of B with c\ c~, c % , etc., as centers and the circles laid 
through 1, 2. 3, etc. Points o, o\ o 2 , o 3 , etc., connected with a 
curve give the face of the tooth ; in like manner the flank is 
obtained. 

In this manner the form of tooth is obtained, when the 
difference of teeth in gear and pinion is less than 15, with the 
exception that the diameter of describing circle B 



- 4^') 



where P = diametral pitch, N and n number of teeth in gears. 

The distances of the tooth above and below the pitch line 
as well as the thickness / are determined as in regular spur 
gears by the pitch, except when the difference in gear and 
pinion is very small, where we obtain a short tooth, as in Figs. 
13 and 14. In such a case the height of tooth is arbitrary and 
onl3 T conditioned by the curve. In internal gears it is best to 
allow more clearance at bottom of tooth than in ordinary spur 
gears. 



29 Teeth 




8 P. 



II 



Fig. 13. 



42 T. 




In a construction of this kind it is suggested to draw the 
tooth outline many times full size and reduce by photography. 
An equally multiplied line A B will help in reducing. 



PROVIDENCE, R. I. 



61 




62 BROWN & SHARPE MFG. CO. 

Seco7id Method. — The difference between gear and pinion 
being very small, it is sometimes desirable to obtain a smooth 
action by avoiding what is termed the "friction of approach- 
ing action."* This is done, the pinion driving, by giving gear 
only flanks, Fig. 15, and the gear driving, by giving gear only 
faces, Fig. 16. In both these cases we have but one describ- 
ing circle, whose diameter is equal to the difference of the two 
pitch diameters. The construction of the curve is precisely 
the same as described under A. The describing circle has 
been divided into 24 parts simply for the sake of greater 
accuracy. 



PART B.— INTERNAL BEVEL GEARS. 

(Fig- 170 

The pitch surfaces of bevel gears are cones whose apexes 
are at a common point, rolling upon each other. The tooth 
forms for any given pair of bevel gears are the same as for a 
pair of spur gears (of same pitch) whose pitch radii are equal to 
the respective apex distances of the normal cones (i. e., cones 
whose elements are perpendicular upon the elements of the 
bevel gear pitch cones). (Compare Fig 19, page 68-) 

The same is true of internal bevel gears, with the modifica- 
tion that here one of the pitch cones rolls inside of the other. 
The spur gears to whose tooth forms the forms of the bevel 
gear teeth correspond, resolve themselves into internal spur 
gears (Fig. 17). The problem is now to be solved as indicated 
in the first part of this chapter. 



* McCord, Kinematics, pages 107, 108. 



PROVIDENCE, R. I. 



63 



8 P. 
Gear 40 Teeth 
Pinion "40 Teeth 




Fig. 17. 



64 BROWN & SHARPE MFG. CO. 



CHAPTER VII. 

GEAR PATTERNS. 

(Fig. 18.) 

To place in bevel gears the best iron where it belongs, the 
tooth side of the pattern should always be in the nowel, no 
matter of what shape the hubs are. 

Hubs, if short, may be left solid on web ; if long they should 
be made loose. A long hub should go on a tapering arbor, to 
prevent tipping in the sand. i c taper for draft on hubs when 
loose, and 3° when solid is considered sufficient. 

Coreprints as a rule are made separate, partly to allow the 
pattern to be turned on an arbor, partly for convenience, 
should it be desirable to use different sizes. 

Put rap- and draw-holes as near to center as possible. 
Referring to Fig. 18, make L = D for D from y A " to i%" t or 
even more, should hubs be very long. Otherwise if D is more 
than 1 *4" leave L = i%". 

Iron pattern before using should be marked, rusted and 
waxed. 

Shrinkage — For cast-iron, J 5" per foot. 
For brass, j^" " 

Cast-iron gears, especially arm gears, do not always shrink 
%" per foot. In making iron patterns the following allow- 
ances have been found useful : 

Up to 12" diameter allow no shrink. 
From 12" to 18" " " JA regular shrink. 

" 18" tO 24" " " V2 " 

" 24" to 48" " " 2 A 

Above 48" " " .10' 

for cast-iron. 



u u 



PROVIDENCE, R. 1. 



65 




66 



BROWN & SHARPE MFG. CO. 



• If in gears the teeth are to be cast, the tooth thickness / in 
the pattern is made smaller than called for by the pitch, to avoid 
binding of the teeth when cast. No definite rule can be given, 
as the practice varies on this point. For the different diam- 
etral pitches we would advise making / smaller by an amount 
expressed in inches, as given in the following table : 



Diam. Pitch. 


Amount t 
is Smaller. 


Diam, Pitch. 


Amount t 
is Smaller. 


16 

12 
IO 

8 
6 


.oio" 
.012" 

.014" 
.016'' 

-ci8 ? ' 


5 
4 
~> 
2 

I 


.020" 
.022" 
.026" 
•03c" 
.040" 



PROVIDENCE, R. I. 



67 



CHAPTER VIII. 



DIMENSIONS AND FORM FOR BEVEL GEAR 

CUTTERS. 

(Fig. 19.) 

The data needed to determine the form and thickness of a 
bevel gear cutter are the following : 
P — pitch. 

N = number of teeth in large gear. 
n— number of teeth in small gear. 
F = length of face of tooth, measured on pitch line, 
After having laid out a diagram of the pitch cones a b c and 
a bf, and laid off the width of face, the problem resolves itself 
into two parts : 

Part I. — Determine Proper Curve for Cutter. 

It will be remembered that in the involute system of cutters 
(the only one used for bevel gears that are cut with rotary 
cutter), a set of eight different cutters is made for each 
pitch, numbering from No. 1 to No. 8, and cutting from 
a rack to 12 teeth. Each number represents the form of 
a cutter suitable to cut the indicated number of teeth. For 
instance, No. 4 cutter (No. 4 curve) will cut 26 to 34 teeth. 
In order to find the curve to be used for gear and pinion 
we simply construct the normal pitch cones by erecting 
the perpendicular p q through b, Fig. 19. We now measure the 
lines b q and b p, and taking them as radii, multiplying each by 
2 and P we obtain a number of teeth for which cutters of 
proper curves may be selected. From example we have : 

Gear : b q — 9^" ; 2 X P X 9.75 = 97 T No. 2 curve. 
Pinion: b p — 3^" ; 2 X P X 3.5 = 35 T No. 3 curve. 

The eight cutters which are made in the involute system 
for each pitch are as follows : 

No. 1 will cut wheels from 135 teeth to a rack. 



" 2 ' 


( a 


a 


55 


a 


a 


134 teeth 


" 3 


1 u 


a 


35 


a 


a 


54 " 


" 4 


( a 


a 


26 


11 


a 


34 " 


" 5 


t a 


a 


21 


a 


a 


25 " 


" 6 


1 u 


it 


17 


a 


a 


20 " 


" 7 


1 a 


a 


14 


u 


a 


16 " 


" 8 


i u 


it 


12 


a 


a 


13 " 



68 



BROWX & SHARPE MFG. CO. 




PROVIDENCE, R. I. 69 

Part II. — Determine Thickness of Cutter. 

It is very evident that a bevel gear cutter cannot be thicker 
than the width of the space at small end of tooth ; the practice 
is to make cutter .005" thinner. Theoretically the cutting angle 
(h) is equal to pitch angle less angle of bottom (or h = a — ft'). 
Practically, however, better results are obtained by making 
h = a — fi (substituting angle of top for angle of bottom), and 
in calculating the depth at small end, to add the full clearance 
(/) to the obtained working depth, giving equal amount of 
clearance at large and small end. This is done to obtain a 
tooth thinner at the top and more curved. As the small end 
of tooth determines the thickness of cutter, we shall have to 
find the tooth part values at small end. From the diagram it 
will be seen that the values at large end are to those at small 
end as their respective apex distances (a b and a I). The 
numerical values of these can be taken' from the diagram and 
the quotient of the larger in the smaller is the constant where- 
with to multiply the tooth values at large end, to obtain those 
at small end. In our example we find : 

. ~~ — = .6^ = constant t? t> , 

afr=c.8 For 5 P we have : 



2057 
1310 

°3£4 

1624 

1310 



/=. 3 i 4 i f = 

S = .2COO / = 

/=.o 3 i4 / = 

^+/=- 2 3 I 4 /+/ = 

D"+/=.43i4- s= = _ 

D'"+/=.2934 

From the foregoing it is evident that a spur gear cutter 
could not be used, since a bevel gear cutter must be thinner. 

If in gears of more than 30 teeth the faces are proportion- 
ately long, we select a cutter whose curve corresponds to the 
midway section of the tooth. The curve of the cutter is found 
by the method explained in Part I. of this Chapter. 



ro BROWN & SHARPE MFG. CO. 



CHAPTER IX. 

DIRECTIONS FOR CUTTING BEVEL GEARS 
WITH ROTARY CUTTER. 

(Fig. 20.) 

In order to obtain good results, the gear blanks must be of 
the right size and form. The following sizes for each end of 
the tooth must be given the workman : 

Total depth of tooth. 
Thickness of tooth at pitch line. 
Height of tooth above pitch line. 

These sizes are obtained as explained in Chapter VIII. 
The workman must further know the cutting angle (see 
formula on page 13 and compare Chapter VIII.), and be pro- 
vided with the proper tools with which to measure teeth, etc. 

In cutting a gear on a universal milling machine the opera- 
tions and adjustments of the machine are as follows : 

1. Set spiral bed to zero line. 

2. Set cutter central with spiral head spindle. 

3. Set spiral head to the proper cutting angle. 

4. Set the index on head for the number of teeth to be cut, 
leaving the sector on the straight or numbered row of holes, 
and set the pointer (or in some machines the dial) on cross-feed 
screw of milling machine to zero line. 

5. As a matter of precaution, mark the depth to be cut for 
large and small end of tooth on their respective places. 

6. Cut two or three teeth in blank to conform with these 
marks in depth. The teeth will now be too thick on both their 
pitch circles. 

7. Set the cutter off the center by moving the saddle to or 
from the frame of the machine by means of the cross-feed 
screw, measuring the advance on dial of same. The saddle 
must not be moved further than what to good judgment 



PROVIDENCE, R. I. 



7* 




Fig. 20. 



72 BROWN & SHARPE MFG. CO. 

appears as not excessive ; at the same time bearing in mind 
that an equal amount of stock is to be taken off each side of 
tooth. 

8. Rotate the gear in the opposite direction from which the 
saddle is moved off the center, and trim the sides of teeth (A) 
(Fig. 20.) 

9. Then move the saddle the same distance on the opposite 
side of center and rotate the gear an equal amount in the 
opposite direction and trim the other sides of teeth (C). 

10. If the teeth are still too thick at large end E, move the 
saddle further off the center and repeat the operation, bearing 
in mind that the gear must be rotated and the saddle moved 
an equal amount each way from their respective zero settings. 

It is generally necessary to file the sides of teeth above the 
pitch line more or less on the small ends of teeth, as indicated 
by dotted lines F F. This applies to pinions of less than 30 
teeth. 

For gears of coarser pitch than 5 diametral it is best to 
make one cut around before attempting to obtain the tooth 
thickness. 

The formulas for obtaining the dimensions and angles of 
gear blanks are given in Chapter III. 



PROVIDENCE, R. I. 



73 



CHAPTER X. 

THE INDEXING OF ANY WHOLE OR FRAC- 
TIONAL NUMBER. 

(Fig. 21 ) 




In indexing on a machine the question simply is : How- 
many divisions of the machine index have to be advanced to 
advance a unit division of the number required. To which 
is the 

divisions of machine index 



answer 



number to be indexed 



Suppose the number of divisions in index wheel of machine 
to be 216. 



Example I. — Index 72. 

Answer: 216 

72 



3 (3 turns of worm). 



74 BROWN & SHARPE MFG. CO. 

Example II. — Index 123. 

— = 1 + .93 
123 123 

If now we should put on worm shaft a change gear having 
123 teeth, give the worm shaft, Fig. 21, one turn, and in addi- 
tion thereto advance 93 teeth of the change gear (to give the 
fractional turn), we would have indexed correctly one unit of 
the given number, and so solved the problem. Should we not 
have change gear 123 we may try those on hand. The ques- 
tion then is : How many teeth (x) of the gear on hand (for 
instance 82) must we advance to obtain a result equal to the 
one when advancing 93 teeth of the 123 tooth gear ? We have : 

-^- = -- where x = 62 
123 82 

Example III. — Index 365, change gear 147. 

= -^- where j = 87 — -2- 

3 6 5 J 47 3 6 5 

Here 147 is the change gear on hand. In indexing for a unit 
of 365 we advance87teeth of our 147 tooth gear. It is evident 
that in so doing we advance too fast and will have indexed 
three teeth of our change gear too many when the circle is 
completed. To avoid having this error show in its total amount 
between the last and the first division, we can distribute the 
error by dropping one tooth at a time at three even intervals. 

Example IV. — Index 190. 

216 26 

Yqo ~ Too Change gear on hand 88 T 

— = — where j = 12 + 

190 88 190 

To distribute the error in this case we advance one additional 
tooth ot a time of the change gear at eight even intervals. 

Example V. — Index 117.3913. 

216 _ ^86087 



H7-39 T 3 ll 739 1 3 

This example is in nowise different from the preceding 
ones, except that the fraction is expressed in large numbers. 
This fraction we can reduce to lower approximate values, 
which for practical purposes are accurate enough. This is 
done by the method of continued fractions. [For an explana- 



PROVIDENCE, R. I. 75 

tion of this method we refer to our " Practical Treatise on 

Gearing."] 

986087 







"739 x 3 


986087) 


I I 73913 
986087 


(1 




187826) 986087 (5 






939 J 30 






46957) 187826 (3 
140871 






46955) 46957 (1 
46955 






2) 46955 (23477 
46954 






j) 2 (2 
2 



"739*3 



1 + 1 

5 + i 

' 3 + i 



1 + 1 



23477 + £ 
2 



r=3 1 23477 



<? = 1 b = 5 </= 16 21 493033 986087 
« l =i £' = 6 ^/ n = 19 25 586944 n739 T 3 

Note.. — Find the first two fractions by reduction = - and : — — z ', the 

11 1 + 1 6 

5 

others are then found by the rule \ c ■" a ~ 

J ! b y c + a 1 ^d 1 



The fraction | } is a good approximation; putting therefore 
a change gear of 25 teeth on worm shaft, we advance (beside 
the one full turn) 21 teeth to index our unit. 

Of course, in using any but the correct fraction we have an 
error every time we index a division ; so that when indexed 
around the whole circle, we have multiplied this error by the 
number of divisions. 

In the present example this error is evidently equal to the 

difference between the correct and the approximate fraction 

used. Reducing both common fractions to decimal fractions 

we have : 

9S6087 Q > 

.<■ — = .84000006 

n739 r 3 



M 

1 2I 

— =.84000000 



.uoooooo6 = error in each division. 






v :: 



76 BROWN & SHARPE MFG. CO. 

.00000006 X 1 1 7.3913 = .00000704348 total error in complete 
circle. This error is expressed in parts of a unit division. (To 
find this error expressed in inches, multiply it by the distance 
between two divisions, measured on the circle.) In this case 
the approximate fraction being smaller than the correct one, 
in indexing the whole circle we fall short .00000704348 of a 
division. 



Example VI. 


— Index 15.708 




216 , 11706 

■ = 13 + — 1 -~ 




15.708 15708 




11796 _ 983 




15708 1309 




983 ) 1309 ( 1 




983 




326) 983 (3 




978 




5) 326 (65 




30 




26 




25 




1) 5 (5 




5 









983 _, 




1309 r^-r 




3+i 




65 + 1 




5 




1 3 65 5 


/ 


1 3 196 983 




1 4 261 1309 



In using the approximation JJ-J the error for each division 
(found as above) will be .000002927, for the whole circle 
.0000460. In this case, the approximation being larger than 
the correct fraction, we overreach the circle by the error. 



PROVIDENCE, R. I. 



77 



CHAPTER XI. 

THE GEARING OF LATHES FOR SCREW 

CUTTING. 

(Figs. 22, 23.) 

The problem of cutting a screw on a lathe resolves itself into 
connecting the lathe spindle with the lead screw by a train of 
gears in such a manner that the carriage (which is actuated by 



^vrvrv-TL/iy^ 




Simple Gearing. 

Fig. 22. 



V ofC. 



78 



BROWN & SHARPK MFG. CO. 



the lead screw) advances just, one inch, cr some definite dis- 
tance, while the lathe spindle makes a number of revolutions 
equal to the number of threads lo be cut per inch. 

The lead screw has, with the exception of a very few cases, 
always a single thread, and to advance the carriage one inch it 
therefore makes a number of revolutions equal to its number 




Compound Gearing- 

Fig. 23. 



of threads per inch. Should the lead screw have double 
thread, it will, to accomplish the same result, make a number 
of revolutions equal to half its number of threads per inch. It 
follows that we must know in the first place the number of 
threads per inch on lead screw. 



PROVIDKNCE, R. I. 79 

It ought to be clearly understood that one or more inter- 
mediate gears, which simply transmit the motion received from 
one gear to another, in no wise alter the ultimate ratio of a 
train of gearing. An even number of intermediate gears 
simply change the direction of rotation, an odd number do not 
alter it. 

The gearing of a lathe to solve a problem in screw cutting 
can be accomplished by 

A. Simple gearing. 

B. Compound gearing. 

Referring to the diagrams, Figs. 22 and 23, we have in Fig. 
22 a case of simple, and in Fig. 23 a case of compound gear- 
ing. 

In simple gearing the motion from gear E is transmitted 
either directly to gear Ron lead screw or through the interme- 
diate F. In compound gearing the motion of E is transmitted 
through two gears (G and H) keyed together, revolving on the 
same stud n, by which we can change the velocity ratio of the 
motion while transmitting it from E to R. With these four 
variables E, G, H, R, we are enabled to have a wider range of 
changes than in simple gearing. 

B and C, being intermediate gears, are not to be considered. 
If, as is generally the case, gear A equals gear D, we disregard 
them both, simply remembering that gear E (being fast on 
same shaft with D) makes as many revolutions as the spindle. 
Sometimes gear D is twice as large as gear A, then, still con- 
sidering gear E as making as many revolutions as the spindle, 
we deal with the lead screw as having twice as many threads 
per inch as it measures. 



SIMPLE GEARING, 

Let there be : the number of teeth in the different gears 
expressed by their respective letters, as per Fig. 22, and 

s = threads per inch to be cut, 
L — threads per inch on lead screw ; then 
1. s _ R 

L " E 



SO BROWN & SHARPE MFG. CO. 

If now one of the two gears E and R is selected, the other 
will be : 



L 

The two gears may be found by making 

R = 
E = 



P _P | > where/ may be any number. 



3. The above holds good when a fractional thread is to be 
cut, but if the fraction is expressed in large numbers, as, for 
instance, j- = 2.833 C 2 ^mr)» we nrst reduce this fraction (y^ 3 -^) t0 
lower approximate values by the process of continued fraction 
(see pages 73 and 74). 



833 . ICOO 

833 

167) 


(I 

8l3 4 

66S 

165) 167 (1 
165 

2) 165 (82 
16 

5 
4_ 

1)2(2 
2 






I 


4 I 82 


2 


I 
I 


A JL ill 

5 6 497 


833 
1000 



— = .8^^ (nearly) and s = 2^ 
u 6 

If in this case L = 4, and we select E = 48, then, since 
R = fJE R = 34 



COMPOUND GEARING. 

4. In a lathe geared compound for cutting a screw the 
product of the drivers (E and H, Fig. 2^) multiplied by the num- 
ber of threads per inch to be cut must equal the product of the 
driven (G and I\) multiplied by the number of threads on lead 
screw. This is expressed by 

F H s 
E . H . s r = G . R . L or - f " / = 1 



PROVIDENCE, R. I. 8 1 

If three of the gears E, H, G, R have been selected, the 
fourth one would be either 



GRL 





H s 


H - 


GRL 




E s 


G- 


E H s 




K L 


R = 


E H s 
G L 





R G L 



or 



or 



or 



= L 
E H 



/J^GX 
VL.E.H/ 



If a fractional thread is to be cut, as under " 3," we reduce 
the fraction to lower approximate values. 

Example.— Gear for 5.2327 threads per inch, lead screw is 
6 threads. 

2^27 
.2327 = -^—L 
IOOOO 

2327) iooco (4 
93o8 
692) 2327 (3 
2076 

251) 692 (2 
502 

190) 251 (1 
190 

~6t) 19c (3 

183 

7) 61 (8 
56 

5)7(i 
5_ 

2; 5 (2 
4 

1)2 (2 
2 



10 37 3° 6 343 99 2 2 3 2 7 



4 13 3° 43 x 59 !3!5 J 474 4263 10000 

10 /- 1 \ j IO 

— = .2327 (nearly) and 5.2327 = 5 — 

43 '43 

Selecting E = 43, H = 52, R = 50, and 

G = we have G = 3£ — - xJJ = ?q. 

R . L 50 . 6 ° y 



82 BROWN & SHARPE MFG. CO. 

5. The examples so far given all deal with single thread. 
The pitch of a screw is the distance from center of one thread to 
the center of the next. The lead of a screw is the advance for 
each complete revolution. In a single thread screw the pitch 
is equal to the lead, while in a double thread screw the pitch 
is equal to one-half the lead ; in a triple thread screw equal to 
one-third the lead, etc. 

If we have to gear a lathe for a many-threaded screw 
(double, triple, quadruple, etc.), we simply ascertain the lead, 
and deal with the lead as we would with the pitch in a single 
thread screw, i. <?., we divide one inch by it, to obtain the num- 
ber of threads for which we have to eear our lathe. 



Example. — Gear for double thread screw, lead = .4654. 
Number of threads per inch to be geared for is : 

= = 2.1487 

Lead -4654 

Lead screw is four threads per inch. 

As in previous examples, we reduce the fraction .14%? = -f££f6 
to lower approximate values by the process of continued frac- 
tion. 

From the different values received in the usual way we 
select : 

4} = .1487 (nearly) and 2.1487 = 2AJ 

We have therefore : 

S = 24-1 

L = 4 

( E = 74 

Selecting ■> G = 30 

/ H = 40 



R = 



E . H. s _ 74 -40. 2H_. -„ 

3J 



G . L 



Note. — In using any but the original fraction we commit an error. This error 
can be found by reducing the approximate fraction used to a decimal fraction, and 
comparing it with the original fraction. In the above example the original fraction is 

.14S7 and 
H = . I4S64 

Error = .oooc6 inch in lead. 



In cutting a multiple screw, after having cut one 
thread, the question arises how to move the thread tool the 
correct amount for cutting the next thread. 



PROVIDENCE, R. I. 83 

In cutting double, triple, etc., threads, if in simple or com- 
pound gearing the number of teeth in gear E is divisible by 
2, 3, etc., we so divide the teeth ; then leaving the carriage 
at rest we bring gear E out of mesh and move it forward one 
division, whereby the spindle will assume the correct position. 

When E is not divisible we find how many turns (V) of 
gear R are made to each full turn of the spindle. Dividing 
this number by 2 for double, by 3 for triple thread, etc., we 
advance R so many turns and fractions of a turn, being careful 
to leave the spindle at rest. 

For compound gearing : 

G . R 

When the gear D is twice as large as the gear A (ss ex- 
plained in fifth paragraph, page 78.) the formula would be 

y= E. H. 
_ 2 G. R. 

If in simple gearing both E and R are not divisible, one 
remedy would be to gear the lathe compound ; or the face- 
plate may be accurately divided in tw T o, three or more slots, 
and all that is then necessary is to move the dog from one slot 
to another, the carriage remaining stationary. 



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